FREE Download Stu Pacage from website: www.teoclasses.com & www.mathsbsuhag.com Get Solution of These Pacages & Learn b Video Tutorials on www.mathsbsuhag.com SHORT REVISION DIFFERENTIAL EQUATIONS OF FIRST ORDER AND FIRST DEGREE DEFINITIONS:. An equation that involves independent and dependent variables and the derivatives of the dependent variables is called a DIFFERENTIAL EQUATION.. A differential equation is said to be ordinar, if the differential coefficients have reference to a single independent variable onl and it is said to be PARTIAL if there are two or more independent variables. We are concerned with ordinar differential equations onl. eg. u u u = 0 is a partial differential equation. z. Finding the unnown function is called SOLVING OR INTEGRATING the differential equation. The solution of the differential equation is also called its PRIMITIVE, because the differential equation can be regarded as a relation derived from it. 4. The order of a differential equation is the order of the highest differential coefficient occuring in it. 5. The degree of a differential equation which can be written as a polnomial in the derivatives is the degree of the derivative of the highest order occuring in it, after it has been epressed in a form free from radicals & fractions so far as derivatives are concerned, thus the differential equation : p q m m d d ( ) f(, ) m d + (, ) m d +... = 0 is order m & degree p. Note that in the differential equation e + = 0 order is three but degree doesn't appl. 6. FORMATION OF A DIFFERENTIAL EQUATION : If an equation in independent and dependent variables having some arbitrar constant is given, then a differential equation is obtained as follows : Differentiate the given equation w.r.t. the independent variable (sa ) as man times as the number of arbitrar constants in it. Eliminate the arbitrar constants. The eliminant is the required differential equation. Consider forming a differential equation for ² = 4a( + b) where a and b are arbitar constant. Note : A differential equation represents a famil of curves all satisfing some common properties. This can be considered as the geometrical interpretation of the differential equation. 7. GENERAL AND PARTICULAR SOLUTIONS : The solution of a differential equation which contains a number of independent arbitrar constants equal to the order of the differential equation is called the GENERAL SOLUTION (OR COMPLETE INTEGRAL OR COMPLETE PRIMITIVE). A solution obtainable from the general solution b giving particular values to the constants is called a PARTICULAR SOLUTION. Note that the general solution of a differential equation of the n th order contains n & onl n independent arbitrar constants. The arbitrar constants in the solution of a differential equation are said to be independent, when it is impossible to deduce from the solution an equivalent relation containing fewer arbitrar constants. Thus the two arbitrar constants A, B in the equation = A e + B are not independent since the equation can be written as = A e B. e = C e. Similarl the solution = A sin + B cos ( + C) appears to contain three arbitrar constants, but the are reall equivalent to two onl. 8. Elementar Tpes Of First Order & First Degree Differential Equations. TYPE. VARIABLES SEPARABLE : If the differential equation can be epressed as ; f ()d + g() = 0 then this is said to be variable separable tpe. A general solution of this is given b f() d + g() = c ; where c is the arbitrar constant. consider the eample (/d) = e +. e. Note : Sometimes transformation to the polar coordinates facilitates separation of variables. In this connection it is convenient to remember the following differentials. If = r cos ; = r sin then, (i) d + = r dr (ii) d + = dr + r d (iii) d = r d If = r sec & = r tan then d = r dr and d = r sec d. TYPE : = f (a + b + c), b 0. d To solve this, substitute t = a + b + c. Then the equation reduces to separable tpe in the variable t and which can be solved. Teo Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : 0 90 90 7779, 0 9890 5888. page 6 of 5 Successful People Replace the words lie; "wish", "tr" & "should" with "I Will". Ineffective People don't.
FREE Download Stu Pacage from website: www.teoclasses.com & www.mathsbsuhag.com Get Solution of These Pacages & Learn b Video Tutorials on www.mathsbsuhag.com Consider the eample ( + ) d = a. TYPE. HOMOGENEOUS EQUATIONS : A differential equation of the form d = f (, ) (, ) where f (, ) & (, ) are homogeneous functions of &, and of the same degree, is called HOMOGENEOUS. This equation ma also be reduced to the form d = g & is solved b putting = v so that the dependent variable is changed to another variable v, where v is some unnown function, the differential equation is transformed to an equation with variables separable. Consider d + ( ) = 0. TYPE4. EQUATIONS REDUCIBLE TO THE HOMOGENEOUS FORM : If d = a b c a ; where a a b c b a b 0, i.e. a b b then the substitution = u + h, = v + transform this equation to a homogeneous tpe in the new variables u and v where h and are arbitrar constants to be chosen so as to mae the given equation homogeneous which can be solved b the method as given in Tpe. If (i) (ii) a b a b = 0, then a substitution u = a + b transforms the differential equation to an equation with variables separable. and b + a = 0, then a simple cross multiplication and substituting d () for + d & integrating term b term ields the result easil. Consider d = 5 ; d = & 4 6 5 d = 6 5 4 (iii) In an equation of the form : f () d + g () = 0 the variables can be separated b the substitution = v. IMPORTANT NOTE : (a) The function f (, ) is said to be a homogeneous function of degree n if for an real number t ( 0), we have f (t, t) = t n f(, ). For e.g. f(, ) = a / + h /. / + b / is a homogeneous function of degree /. (b) A differential equation of the form = f(, ) is homogeneous if f(, ) is a homogeneous d function of degree zero i.e. f(t, t) = t f(, ) = f(, ). The function f does not depend on & separatel but onl on their ratio or. LINEAR DIFERENTIAL EQUATIONS : A differential equation is said to be linear if the dependent variable & its differential coefficients occur in the first degree onl and are not multiplied together. The nth order linear differential equation is of the form ; n n a 0 () d n d + a () d n d +... + a (). = (). Where a (), a ()... a () are called the n 0 n coefficients of the differential equation. Note that a linear differential equation is alwas of the first degree but ever differental equation of the first degree need not be linear. e.g. the differential equation d d + d = 0 is not linear, though its degree is. TYPE 5. LINEAR DIFFERENTIAL EQUATIONS OF FIRST ORDER : The most general form of a linear differential equations of first order is + P = Q, where P & Q are d functions of. To solve such an equation multipl both sides b e P d. Teo Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : 0 90 90 7779, 0 9890 5888. page 7 of 5 Successful People Replace the words lie; "wish", "tr" & "should" with "I Will". Ineffective People don't.
FREE Download Stu Pacage from website: www.teoclasses.com & www.mathsbsuhag.com Get Solution of These Pacages & Learn b Video Tutorials on www.mathsbsuhag.com NOTE : () The factor e Pd on multipling b which the left hand side of the differential equation becomes the differential coefficient of some function of &, is called integrating factor of the differential equation popularl abbreviated as I. F. () It is ver important to remember that on multipling b the integrating factor, the left hand side becomes the derivative of the product of and the I. F. () Some times a given differential equation becomes linear if we tae as the independent variable and as the dependent variable. e.g. the equation ; ( + + ) d = + can be written as ( + ) d = + + which is a linear differential equation. TYPE6. EQUATIONS REDUCIBLE TO LINEAR FORM : The equation d + p = Q. n where P & Q functions of, is reducible to the linear form b dividing it b n & then substituting n+ = Z. Its solution can be obtained as in Tpe5. Consider the eample ( + ) d =. The equation d + P = Q. n is called BERNOULI S EQUATION. 9. TRAJECTORIES : Suppose we are given the famil of plane curves. (,, a) = 0 depending on a single parameter a. A curve maing at each of its points a fied angle with the curve of the famil passing through that point is called an isogonal trajector of that famil ; if in particular =/, then it is called an orthogonal trajector. Orthogonal trajectories : We set up the differential equation of the given famil of curves. Let it be of the form F (,, ') = 0 The differential equation of the orthogonal trajectories is of the form F,, = 0 The general integral of this equation (,, C) = 0 gives the famil of orthogonal trajectories. Note : Following eact differentials must be remembered : (i) + d = d() (ii) (iv) (vii) () d d(ln ) (v) d d ln (viii) d d d (iii) = d (ln( + )) (vi) d d tan (i) d d d d ln d d tan d d d ln (i) d (ii) d e e d e (iii) d e e e d EXERCISE I [ FORMATION & TYPE & TYPE ] Q. State the order and degree of the following differential equations : 4 d d (i) d t d t = 0 (ii) d t d d Q. Form a differential equation for the famil of curves represented b a² + b² =, where a & b / Successful People Replace the words lie; "wish", "tr" & "should" with "I Will". Ineffective People don't. Teo Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : 0 90 90 7779, 0 9890 5888. page 8 of 5
FREE Download Stu Pacage from website: www.teoclasses.com & www.mathsbsuhag.com Get Solution of These Pacages & Learn b Video Tutorials on www.mathsbsuhag.com are arbitar constants. Q. Obtain the differential equation of the famil of circles + + g + f + c = 0 ; where g, f & c are arbitar constants. Q.4 Form the differential equation of the famil of curves represented b, c ( + c) = ; where c is an arbitrar constant. n (sec tan ) Q.5 d = n (sec tan ) Q.6 ( ²) ( ) d = ( + ) cos cos Q.7 d + = 0 Q.8 a d d d d Q.9 = Q.0 = sin ( + ) + cos ( + )Q. d d d = ( ln ) sin cos Q. It is nown that the deca rate of radium is directl proportional to its quantit at each given instant. Find the law of variation of a mass of radium as a function of time if at t = 0, the mass of the radius was m 0 and during time t 0 % of the original mass of radium deca. Q. d + sin sin Q.4 Sin. d =. ln if = e, when = Q.5 e (/d) = + given that when = 0, = Q.6 A normal is drawn at a point P(, ) of a curve. It meets the ais at Q. If PQ is of constant length, then show that the differential equation describing such curves is, d = ±. Find the equation of such a curve passing through (0, ). Q.7 Find the curve for which the sum of the lengths of the tangent and subtangent at an of its point is proportional to the product of the coordinates of the point of tangenc, the proportionalit factor is equal to. Q.8 Obtain the differential equation associated with the primitive, = c e + c e + c e, where c, c, c are arbitrar constants. Q.9 A curve is such that the length of the polar radius of an point on the curve is equal to the length of the tangent drawn at this point. Form the differential equation and solve it to find the equation of the curve. Q.0 Find the curve = f () where f () 0, f (0) = 0, bounding a curvilinear trapezoid with the base [0, ] whose area is proportional to (n + ) th power of f (). It is nown that f () =. EXERCISE II [ TYPE & TYPE4] Q. d = Q. Find the equation of a curve such that the projection of its ordinate upon the normal is equal to its abscissa. Q. The light ras emanating from a point source situated at origin when reflected from the mirror of a search light are reflected as beam parallel to the ais. Show that the surface is parabolic, b first forming the differential equation and then solving it. Q.4 The perpendicular from the origin to the tangent at an point on a curve is equal to the abscissa of the point of contact. Find the equation of the curve satisfing the above condition and which passes through (, ). Q.5 Find the equation of the curve intersecting with the - ais at the point = and for which the length of the subnormal at an point of the curve is equal to the arthemetic mean of the co-ordinates of this point ( ) ( + ) =. Q.6 Use the substitution = a to reduce the equation. d + + = 0 to homogeneous form and hence solve it. Q.7 Find the isogonal trajectories for the famil of rectangular hperbolas = a which maes with it an angle of 45. Q.8 ( ) d = ( ) f (, ) Q.9 Show that ever homogeneous differential equation of the form = where f and g are d g(, ) homogeneous function of the same degree can be converted into variable separable b the substitution = r cos and = r sin. Q.0 cos sin sin cos d = 0 Teo Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : 0 90 90 7779, 0 9890 5888. page 9 of 5 Successful People Replace the words lie; "wish", "tr" & "should" with "I Will". Ineffective People don't.
FREE Download Stu Pacage from website: www.teoclasses.com & www.mathsbsuhag.com Get Solution of These Pacages & Learn b Video Tutorials on www.mathsbsuhag.com Q. Find the curve for which an tangent intersects the ais at the point equidistant from the point of tangenc and the origin. Q. ( ) = ( + + ) d Q. d = d Q.4 d = 5 Q.5 d = Q.6 d = ( ) d cos ( cos 7 sin ) Q.7 + ( ) d sin ( sin 7 cos 7) = 0 Q.8 Show that (4 + + ) d + ( + + ) = 0 represents a hperbola having an asmptotes, + = 0 & + + = 0. Q.9 If the normal drawn to a curve at an point P intersects the -ais at G and the perpendicular from P on the -ais meets at N, such that the sum of the lengths of PG and NG is proportional to the abscissa of the point P, the constant of proportionalit being. Form the differential equation and solve it to show that the equation of the curve is, = c or = c, where c is an arbitrar constant. Q.0 Show that the curve such that the distance between the origin and the tangent at an arbitrar point is equal to the distance between the origin and the normal at the same point, = c e tan EXERCISE III [ TYPE5 & TYPE6 ] Q. ( + tan ) = sin d Q. Show that the equation of the curve whose slope at an point is equal to + and which pass through the origin is = (e ). d Q. d + = Q.4 ( ²) + = ( ²)/ ( ) d Q.5 Find the curve such that the area of the trapezium formed b the coordinate aes, ordinate of an arbitrar point & the tangent at this point equals half the square of its abscissa. Q.6 ( ) d ( ) = ( ) Q.7 ( + + ²) d + ( + ) = 0 Q.8 Find the curve possessing the propert that the intercept, the tangent at an point of a curve cuts off on the ais is equal to the square of the abscissa of the point of tangenc. Q.9 sin + = cos Q.0 (² + ) d d = ( ²) +. ln Q. = ² cosec d Q. ( + ²) d = (tan ) Q. Find the curve such that the area of the rectangle constructed on the abscissa of an point and the initial ordinate of the tangent at this point is equal to a. (Initial ordinate means intercept of the tangent). Q.4 Let the function ln f() is defined where f() eists for & is fied positive real number, prove that if d d (. f ()) f () then f() A where A is independent of. Q.5 Find the differentiable function which satisfies the equation f () = f (t) tan t dt tan(t ) dt 0 0 where, Q.6 D = b( + ²D) Q.7 Integrate ( + ²) + 4² = 0 and obtain the cubic curve satisfing this equation and d passing through the origin. Q.8 If & be solutions of the differential equation + P = Q, where P & Q are functions of d alone, and = z, then prove that Q d z = + a e, 'a' being an arbitrar constant. Q.9 d + ln = (ln ) Q.0 d + = ²e²/. sin Q. d sec = tan Q. d = 4 cos Q. ( + e ) d e = 0 Teo Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : 0 90 90 7779, 0 9890 5888. page 0 of 5 Successful People Replace the words lie; "wish", "tr" & "should" with "I Will". Ineffective People don't.
FREE Download Stu Pacage from website: www.teoclasses.com & www.mathsbsuhag.com Get Solution of These Pacages & Learn b Video Tutorials on www.mathsbsuhag.com Q.4 Find the curve for which the area of the triangle formed b the ais, the tangent line and radius vector of the point of tangenc is equal to a. Q.5 A tan contains 00 litres of fresh water. A solution containing gm/litre of soluble lawn fertilizer runs into the tan at the rate of lit/min, and the miture is pumped out of the tan at the rate of litres/min. Find the time when the amount of fertilizer in the tan is maimum. EXERCISE IV (GENERAL CHANGE OF VARIABLE BY A SUITABLE SUBSTITUTION) Q. ( ²) d + = 0 Q. ( + + ) d + = 0 Q. Q 6. Q 9. Q. Q. d + ln = e Q 4. d tan = ( + ) e sec Q 5. d = e ( ) 0 Q 7. d d d = Q 8. ( + ) d = ( ) d = e (e e ) Q 0. sin = cos (sin ) EXERCISE V(MISCELLANEOUS) d ln = sin. (cos ) ln, being bounded when +. d = + d given =, where = 0 0 Q. Given two curves = f() passing through the points (0, ) & = Successful People Replace the words lie; "wish", "tr" & "should" with "I Will". Ineffective People don't. f(t) dt passing through the points (0, /). The tangents drawn to both curves at the points with equal abscissas intersect on the ais. Find the curve f(). Q.4 Consider the differential equation + P() = Q() d (i) If two particular solutions of given equation u() and v() are nown, find the general solution of the same equation in terms of u() and v(). (ii) If and are constants such that the linear combinations u() + v() is a solution of the given equation, find the relation between and. (iii) If w() is the third particular solution different from u() and v() then find the ratio v ( ) u ( ). w( ) u( ) Q.5 d = + Q.6 Find the curve which passes through the point (, 0) such that the segment of the tangent between the point of tangenc & the ais has a constant length equal to. d d d Q.7 + d + = 0 Q.8, given that = when = Q.9 Find the equation of the curve passing through the orgin if the middle point of the segment of its normal from an point of the curve to the -ais lies on the parabola =. Q.0 Find the continuous function which satisfies the relation, t f ( t)dt = 0 0 f (t) dt + sin + cos, for all real number. Q. ( + + a ) d d + ( + a ) = 0 Q.( ) d + d = 0 Q. + cos () sin () + d d { sin ()} = 0. Q.4 Find the integral curve of the differential equation, ( l n ). + = 0 which passes through d,. e Q.5 Find all the curves possessing the following propert ; the segment of the tangent between the point of tangenc & the -ais is bisected at the point of intersection with the -ais. Q.6 ( d + ) ( d + ) = 0 Teo Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : 0 90 90 7779, 0 9890 5888. page of 5
FREE Download Stu Pacage from website: www.teoclasses.com & www.mathsbsuhag.com Get Solution of These Pacages & Learn b Video Tutorials on www.mathsbsuhag.com Q.7 A perpendicular drawn from an point P of the curve on the ais meets the ais at A. Length of the perpendicular from A on the tangent line at P is equal to 'a'. If this curve cuts the -ais orthogonall, find the equation to all possible curves, epressing the answer eplicitl. Q.8 Find the orthogonal trajectories for the given famil of curves when 'a' is the parameter. (i) = a (ii) cos = a e (iii) + = a Q.9 A curve passing through (, 0) such that the ratio of the square of the intercept cut b an tangent off the -ais to the subnormal is equal to the ratio of the product of the co-ordinates of the point of tangenc to the product of square of the slope of the tangent and the subtangent at the same point. Determine all such possible curves. Q.0 A & B are two separate reservoirs of water. Capacit of reservoir A is double the capacit of reservoir B. Both the reservoirs are filled completel with water, their inlets are closed and then the water is released simultaneousl from both the reservoirs. The rate of flow of water out of each reservoir at an instant of time is proportional to the quantit of water in the reservoir at that time. One hour after the water is released, the quantit of water in reservoir A is.5 times the quantit of water in reservoir B. After how man hours do both the reservoirs have the same quantit of water? Q. (² + ² 7) d = (² + 8) Q. Find the curve such that the segment of the tangent at an point contained between the -ais and the straight line = a + b is bisected b the point of tangenc. Q. Find the curve such that the ratio of the distance between the normal at an of its point and the origin to the distance between the same normal and the point (a, b) is equal to the constant. Interpret the curve. ( > 0) Q.4 Let f (,, c ) = 0 and f (,, c ) = 0 define two integral curves of a homogeneous first order differential equation. If P and P are respectivel the points of intersection of these curves with an arbitrar line, = m then prove that the slopes of these two curves at P and P are equal. Q.5 Find the curve for which the portion of -ais cut-off between the origin and the tangent varies as cube of the absissa of the point of contact. EXERCISE VI (PROBLEMS ASKED IN JEE & REE) Q. Determine the equation of the curve passing through the origin in the form = f (), which satisfies the differential equation d = sin (0 + 6 ). [ JEE '96, 5 ] Q. Solve the differential equation ; cos d (tan ) = cos4, <, when (/ 6)= /8. 4 cos d sin 0, when () = Q. Solve the diff. equation ; d Q.4 Let u () & v () satisf the differential equations d u d + p () u = f () & dv d + p () v = g () where p (), f () & g () are continuous functions. If u ( ) > v ( ) for some and f () > g () for all >, prove that an point (, ) where > does not satisf the equations = u () & = v (). [ JEE '97, 5 ] Q.5(i) The order of the differential equation whose general solution is given b (ii) = (C + C ) cos( + C ) C 4 e C 5 where C, C, C, C 4, C 5 are arbitrar constants, is (A) 5 (B) 4 (C) (D) A curve C has the propert that if the tangent drawn at an point P on C meets the coordinate aes at A and B, then P is the mid-point of AB. The curve passes through the point (, ). Determine the equation of the curve. Q.6 Solve the differential equation ( + tan) (d ) + = 0 [ REE '98, 6 ] Q.7(a) A soluton of the differential equation, d d + = 0 is : (A) = (B) = (C) = 4 (D) = 4 (b) The differential equation representing the famil of curves, = c c, where c is a positive parameter, is of : (A) order (B) order (C) degree (D) degree 4 (c) A curve passing through the point (, ) has the propert that the perpendicular distance of the origin from the normal at an point P of the curve is equal to the distance of P from the ais. Determine the equation of the curve. [ JEE '99, + + 0, out of 00 ] Teo Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : 0 90 90 7779, 0 9890 5888. page of 5 Successful People Replace the words lie; "wish", "tr" & "should" with "I Will". Ineffective People don't.
FREE Download Stu Pacage from website: www.teoclasses.com & www.mathsbsuhag.com Get Solution of These Pacages & Learn b Video Tutorials on www.mathsbsuhag.com Q.8 Solve the differential equation, ( + 4 + 4) = ( + 4 + ) d. [ REE '99, 6 ] Q.9 A countr has a food deficit of 0 %. Its population grows continuousl at a rate of %. Its annual food production ever ear is 4 % more than that of the last ear. Assuming that the average food requirement per person remains constant, prove that the countr will become self-sufficient in food after ' where ' n ' is the smallest integer bigger than or equal to, n ' ears, n0 n9. [ JEE '000 (Mains) 0 ] n ( 04. ) 0. 0 Q.0 A hemispherical tan of radius metres is initiall full of water and has an outlet of cm cross sectional area at the bottom. The outlet is opened at some instant. The flow through the outlet is according to the law V(t) = 0.6 gh(t), where V(t) and h(t) are respectivel the velocit of the flow through the outlet and the height of water level above the outlet at time t, and g is the acceleration due to gravit. Find the time it taes to empt the tan. [ JEE '00 (Mains) 0 ] Q. Find the equation of the curve which passes through the origin and the tangent to which at ever point (, ) has slope equal to 4. [ REE '00 (Mains) ] Q. Let f(), > 0, be a nonnegative continuous function, and let F() = f( t) dt, > 0. If for some c > 0, 0 f() < cf() for all > 0, then show that f() = 0 for all > 0. [ JEE 00 (Mains) 5 out of 00 ] Q.(a) A right circular cone with radius R and height H contains a liquid which evaporates at a rate proportional to its surface area in contact with air (proportionalit constant = > 0). Find the time after which the cone is empt. dp() (b) If P() = 0 and > P() for all > then prove that P() > 0 for all >. d [JEE 00, (Mains) 4 + 4] sin Q.4(a) If = cos, (0) =, then = d (A) (B) (C) (D) [JEE 004 (Scr.) ] 4 ( ) (b) A curve passes through (, 0) and the slope of tangent at point P (, ) equals. Find the ( ) equation of the curve and area enclosed b the curve and the -ais in the fourth quadrant. Q.5(a) The solution of primitive integral equation ( + ) = d, is = (). If () = and ( 0 ) = e, then 0 is (A) (e ) (B) (e ) (C) e (D) e (b) For the primitive integral equation d + = ; R, > 0, = (), () =, then ( ) is (A) (B) (C) (D) 5 [JEE 005 (Scr.)] (c) If length of tangent at an point on the curve = f () intercepted between the point and the ais is of length. Find the equation of the curve. [JEE 005 (Mains)] Q.6 A tangent drawn to the curve, = f () at P(, ) cuts the -ais at A and B respectivel such that BP : AP = :, given that f () =, then (A) equation of the curve is = 0 (B) equation of curve is d + = 0 d (C) curve passes through (, /8) (C) normal at (, ) is + = 4 [JEE 006, 5] ANSWER KEY EXERCISE I Q. (i) order & degree (ii) order & degree Q. d + d d d = 0 Q. [ + ()²]. ()² = 0 Q 4. ()² = [8() 7] Q 5. ln (sec + tan ) ln (sec + tan ) = c Q 6. ln ( )² = c ² + ² Teo Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : 0 90 90 7779, 0 9890 5888. page of 5 Successful People Replace the words lie; "wish", "tr" & "should" with "I Will". Ineffective People don't.
FREE Download Stu Pacage from website: www.teoclasses.com & www.mathsbsuhag.com Get Solution of These Pacages & Learn b Video Tutorials on www.mathsbsuhag.com Q 7. sec + = c Q 8. = c ( a) ( + a) Q 9. + = c ( ) Q 0. ln tan = + c Q. sin = ² ln + c Q. m = m 0 e t where = ln t0 00 Q. ln tan 4 = c sin Q.4 = e tan(/) Q 5. = ( + ). ln ( + ) + Q 6. + = Q 7. = n c Q 8. d d 6 6 = 0 d d d Q 9. = or = c Q 0. = /n Q. c ( ) / (² + + ²) /6 = ep tan EXERCISE II where ep e c Q. = n, where same sign has to be taen. Q 4. ² + ² = 0 Q 5. ( ) ( + ) = Q 6. ln + a tan a = c, where a = + Q 7. + = c ; = c Q 8. ² ² = c (² + ²)² Q 0. cos = c Q. + = c Q. arctan = ln c Q. ( + ) = c ( ) Q 4. tan + ln c = 0 Q 5. + + 4 Q 7. (cos sin = ce() ) (cos + sin ) 5 = c Q 6. e tan EXERCISE III Q. cot = c + tan Q. = (e ) = c. ( + ) Q. = c + ln tan tan arc Another form is = c + ln Q 4. = c ( ²) + Q 5. = c ± Q 6. ( ) = ( + c)q 7. = c arc tan Q 8. = c Q 9. tan = c + tan Q 0. 4 (² + ) + ( ln) = c Q. = c + ln tan Q. = ce arctan + arc tan Q. = c ± a Q.5 cos Q 6. ( + b) = b + c Q 7. ( + ²) = 4 Q 9. = ln c Q 0. e ²/ = (c + cos) Q. = + (c + ) cot 4 Q. = Sin + c Q. e = c ² Q 4. = c ± a 7 Q 5. 7 minutes 9 Teo Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : 0 90 90 7779, 0 9890 5888. page 4 of 5 Successful People Replace the words lie; "wish", "tr" & "should" with "I Will". Ineffective People don't.
FREE Download Stu Pacage from website: www.teoclasses.com & www.mathsbsuhag.com Get Solution of These Pacages & Learn b Video Tutorials on www.mathsbsuhag.com EXERCISE IV Q. ² + ln a = 0 Q. ² = ² 6 + ce + 4 Q. ln = e ( ) + c Q 4. sin = (e + c) ( + ) Q 5. c² + e = Q 6. = ce ; = c + Q 7. c = + ( + ) n or + c ( + ) ln Q 8. = tan n c Q 9. e = c. ep (e ) + e Q 0. = c sin sin EXERCISE V Q. = sin Q. = e ( e e + ) Q. f () = e Q.5 (i) = u() + K(u() v()) where K is an constant ; (ii) + = ; (iii) constant Q.5 = c 4 Q.6 = ± 4 n Q.7 + tan = c sin Q.8 Q.9 = + e Q.0 f () = e cos 4 Q. (² + ²)² + a² (² ²) = c Q. = + c e Q. (² ² + cos ) = c Q.4 (e + l n + ) = / a / a Q.5 ² = c Q.6 ( ) = c Q.7 = ± a. e e & = ± a Q.8 (i) + = c, (ii) sin = ce, (iii) = c if = and if c / / Q.9 = e ; = e Q.0 T = log 4/ hrs from the start Q. ( ) 5 = c ( + ) Q. a b = c Q. ( + ) + ( + ) a b = c or ( ) + ( ) a b = c both represents a circle. Q.5 + K = c EXERCISE VI Q. = 5tan 4 5 tan Q. = 4 tan 4 tan. cos Q. sin Q.5 (i) C (ii) = ( > 0, > 0) Q.6 e (cos + sin) = e sin + C Q.7 (a) C (b) A, C (c) + = 0 Q.8 = ln (( + ) + 4( + ) + ) Q.0 7 0 5 Q. (a) T = H Q.5 (a) C; (b) A; (c) 5 ln + c g sec. Q. = ( tan ) ( + ) Q.4 (a) C ; (b) =, area = 4 sq. units l n = ± + c Q.6 B, C Teo Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : 0 90 90 7779, 0 9890 5888. page 5 of 5 Successful People Replace the words lie; "wish", "tr" & "should" with "I Will". Ineffective People don't.
FREE Download Stu Pacage from website: www.teoclasses.com & www.mathsbsuhag.com Get Solution of These Pacages & Learn b Video Tutorials on www.mathsbsuhag.com EXERCISE VII Onl one correct option. The degree of differential equation satisfing the relation + = ( ) is : (A) (B) (C) (D) none of these. If p and q are order and degree of differential equation + d d + = cos, then d (A) p < q (B) p = q (C) p > q (D) none of these. The differential equation for all the straight lines which are at a unit distance from the origin is (A) = d d (B) = + d d (C) = + (D) = d d d d 4. The differential equation obtained on eliminating A and B from = = A cos (t) + B sin (t) is (A) + = 0 (B) = 0 (C) = (D) + = 0 5. The differential equation whose solution is ( h) + ( ) = a is (a is a constant) (A) d = a d d (B) d = a d d (C) d = a d d (D) none of these 6. The differential equation of all circles which pass through the origin and whose centres lie on -ais is (A) ( ) = 0 d (C) ( ) = 0 d (B) ( ) + = 0 d (D) ( ) + = 0 d 7. If = e and = 0 when = 5, the value of for = is d e 6 9 (A) e 5 (B) e 6 + (C) (D) log e 6 8. If () = () and () =, then () equals (A) e (B) e (C) e (D) e 9. If integrating factor of ( ) + ( a ) d = 0 is (A) a ( ) (B) ( ) (C) a p. d e, then P is equal to (D) ( ( 0. If = + + + and ( ) = 0, then function is d (A) ( ) / e (B) ( ) / e (C) log e ( + ) (D) +. Integral curve satisfing =, () =, has the slope at the point (, ) of the curve, equal to (A) 5 dv. The solution of + v = g is dt m t m (A) v = ce mg (B) (C) (D) 5 (B) v = c mg t e m (C) v t m e = c. The solution of the differential equation a + = 0 is d mg t ) (D) v e m = c ) mg Teo Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : 0 90 90 7779, 0 9890 5888. page 6 of 5 Successful People Replace the words lie; "wish", "tr" & "should" with "I Will". Ineffective People don't.
FREE Download Stu Pacage from website: www.teoclasses.com & www.mathsbsuhag.com Get Solution of These Pacages & Learn b Video Tutorials on www.mathsbsuhag.com (A) = Ae / (a ) a (B) = Ae / (a ) a (C) = Ae / (a + ) a (D) = Ae / (a ) a Where A is an arbitrar constant. 4. If () and () are two solutions of + () = r() then () + d () is solution of : (A) + f() = 0 d (C) + f() = r() d (B) + f() = r() d (D) + f () = r() d 5. The differential equation of all 'Simple Harmonic Motions' of given period is n d d d d (A) + n = 0 (B) + n = 0 (C) n = 0 (D) dt dt dt dt 6. If ( ) = tan ( / ) ae, a > 0. Then (0), equals (A) a e / (B) ae / (C) a e / (D) a e / 7. The function f() = d d d satisfies the differential equation coscos df df (A) + f() cot = 0 (B) f() cot = 0 (C) d d 8. The solution of the differential equation = is 0 (A) = A + A + A (B) = A + A (C) = A + A (D) none of these 9. The solution of d + e d = 0 is Successful People Replace the words lie; "wish", "tr" & "should" with "I Will". Ineffective People don't. df d + f() = 0 (D) df d + = 0. n f () = 0 (A) + e = C (B) e = 0 (C) + e = C (D) none of these 0. The solution of the differential equation ( sin cos ) d + ( cos sin sin ) = 0 is (A) sin = sin + C (B) sin + sin = C (C) sin + sin = C (D) sin + sin = C One or more than one options correct. The differential equation of the curve for which the initial ordinate of an tangent is equal to the corresponding subnormal (A) is linear (B) is homogeneous (C) has separable variables (D) is none of these. The solution of + 6 = 0 are (A) = C (B) = C (C) log = C+ log (D) = C. The orthogonal trajectories of the sstem of curves d (A) 9 a( + c) = 4 (B) + C = 4. The solution of ( d + ) = is e (A) / = + C / (B) (C) / = + e / (D). d Solve : d. Solve : a / (C) + C = = a/ are / (D) none of these a the solution of an equation which is reducible to linear equation. = + Ce / EXERCISE VIII Teo Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : 0 90 90 7779, 0 9890 5888. page 7 of 5
FREE Download Stu Pacage from website: www.teoclasses.com & www.mathsbsuhag.com Get Solution of These Pacages & Learn b Video Tutorials on www.mathsbsuhag.com (a) + ( + ) d = 0, given that =, when = (b) cos sin sin cos = 0 d. Find the equation of the curve satisfing d and passing through (, ). d d 4. Find the solution of the differential equation = 8 satisfing (0) =, (0) = 0 and d d 8 (0) =. 5. Solve :(i) ( + ) =, > 0 (ii) ( + + ) d + ( + ) = 0 d (iii) = tan sin (iv) ( + ) + = cos d d 6. Solve :(i) ( + e ) d = e (ii) + = 4 d (iii) sin + ( cos + ) d = 0 d 7. Solve the following differential equations. + d = 8. Find the curve = f() where f() 0, f(0) = 0, bounding a curvilinear trapezoid with the base [0, ] whose area is propostinal to (n + ) th power of f(). It is nown that f() = 9. Find the nature of the curve for which the length of the normal at the point P is equal to the radius vector of the point P. 0. A particle, P, starts from origin and moves along positive direction of -ais. Another particle, Q, follows P i.e. it s velocit is alwas directed towards P, in such a wa that the distance between P and Q remains constant. If Q starts from (, 0), find the equation of the path traced b Q. Assume that the start moving at the same instant.. Let c and c be two integral curves of the differential equation d. A line passing through origin meets c at P(, ) and c at Q(, ). If c : = f() and c : = g() prove that f ( ) = g ( ).. Find the integral curve of the differential equation ( ) + = 0 which passes through (, /e). d. Show that the integral curves of the equation ( ) + = a are ellipses and hperbolas, with the d centres at the point (0, a) and the aes parallel to the co-ordinate aes, each curve having one constant ais whose length is equal to. 4. If & be solutions of the differential equation + P = Q, where P & Q are functions of alone, d and = z, then prove that z = +,'a' being an arbitrar constant. 5. Find the curve for which the sum of the lengths of the tangent and subtangent at an of its point is proportional to the product of the coordinates of the point of tangenc, the proportionalit factor is equal to. 6. Find all the curves possessing the following propert; the segment of the tangent between the point of tangenc & the ais is bisected at the point of intersection with the ais. 7. A curve passing through (, 0) such that the ratio of the square of the intercept cut b an tangent off the ais to the subnormal is equal to the ratio of the product of the coordinates of the point of tangenc to the product of square of the slope of the tangent and the subtangent at the same point. Determine all such possible curves. 8. A & B are two separate reservoirs of water. Capacit of reservoir A is double the capacit of reservoir B. Both the reservoirs are filled completel with water, their inlets are closed and then the water is released simultaneousl from both the reservoirs. The rate of flow of water out of each reservoir at an instant of time is proportional to the quantit of water in the reservoir at that time. One hour after the water is released, the quantit of water in reservoir A is.5 times the quantit of water in reservoir B. After how man hours do both the reservoirs have the same quantit of water? 9. A curv e = f() passes through t he point P (,). The normal to the curv e at P is ; a ( ) + ( ) = 0. If the slope of the tangent at an point on the curve is proportional to the ordinate of the point, determine the equation of the curve. Also obtain the area bounded b the ais, the curve & the normal to the curve at P. [IIT - 996, 5 ] a e Successful People Replace the words lie; "wish", "tr" & "should" with "I Will". Ineffective People don't. Q d Teo Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : 0 90 90 7779, 0 9890 5888. page 8 of 5
FREE Download Stu Pacage from website: www.teoclasses.com & www.mathsbsuhag.com Get Solution of These Pacages & Learn b Video Tutorials on www.mathsbsuhag.com 0. Let u () & v () satisf the differential equations d u d + p () u = f () & dv d + p () v = g () where p (), f () & g () are continuous functions. If u ( ) > v ( ) for some and f () > g () for all >, prove that an point (, ) where > does not satisf the equations = u () & = v ().. A curve passing through the point (, ) has the propert that the perpendicular distance of the origin from the normal at an point P of the curve is equal to the distance of P from the ais. Determine the equation of the curve. [IIT - 999, 0 ]. A countr has a food deficit of 0 %. Its population grows continuousl at a rate of % per ear. Its annual food production ever ear is 4 % more than that of the last ear. Assuming that the average food requirement per person remains constant, prove that the countr will become selfsufficient in food after ' n ' ears, where ' n ' is the smallest integer bigger than or equal to, n0 n9. n ( 04. ) 0. 0 [IIT - 000 (Mains) 0 ]. An inverted cone of height H and radius R is pointed at bottom. It is filled with a volatile liquid completel. If the rate of evaporation is directl proportional to the surface area of the liquid in contact with air (constant of proportionalit > 0), find the time in which whole liquid evaporates. [IIT - 00 (Mains) 4] EXERCISE VII. A. C. C 4. C 5. B 6. A 7. C 8. B 9. D 0. B. A. A. A 4. C 5. B 6. C 7. A 8. A 9. A 0. A. AB. ACD. ABC 4. ABD EXERCISE VIII. c( ). (a) = + (b) cos = c. + = 0 4. 64 = (e 8 8) + 7 5. (i) = + c (ii) = c arc tan (iii) = cos + c sec (iv) ( + ) = c + sin. 6. (i) e = + c (ii) = + c (iii) sin = + c 7. ( + ) = 6 6 + 5 5 + 4 4 + c 8. = /n 9. Rectangular hperbola or circle. 0. = n n ( 4 ) 4. (e + n + ) = 5. = n c 6. ² = c 7. = e / 8. T = log 4/ hrs from the start ; = e 9. e a() a a e a, sq. unit. (c) + = 0. t = H/ For 9 Years Que. from IIT-JEE(Advanced) & 5 Years Que. from AIEEE (JEE Main) we distributed a boo in class room / Teo Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : 0 90 90 7779, 0 9890 5888. page 9 of 5 Successful People Replace the words lie; "wish", "tr" & "should" with "I Will". Ineffective People don't.