Differential Equation. Introduction Booklet - I With the help of this booklet you can easily follow the concepts of differential equations.
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1 Differential Equation Introduction Booklet - I With the help of this booklet you can easily follow the concepts of differential equations. Dr. Ajay Kumar Tiwari 1/16/019
2 Definition The equation which contains the derivative of y = f() i.e. /, independent variable and dependent variable y is known as differential equation. Formation of differential equation Differential equation is formed by eliminating arbitrary constants from the given function. Eample 1 Find the differential equation of all straight lines represented by y = m + c. Differentiating the given equation w.r.t. we get m since in the given equation there are two d y arbitrary constants the again differentiating we get 0. This is the differential equation of second order. hence order of differential equation is highest power of derivative. Concept builder Find the equation of all straight lines having a constant slope m and making a positive intercept c by using calculus. d y Since slope is constant hence 0 integrating both the w.r.t. we get k given that slope is m hence k = m, we get m, again integrating we get y = m + p, ATQ line is passing through (0, c) we get p = c so the required equation of line is y = m + c.
3 Eample Form the differential equation of family of lines situated at a constant distance p from the origin. Let the line is y = m + c, ATQ the length of perpendicular from origin is c p 1 m hence the line is y m p 1 m..(1) differentiating w.r.t. we get m substituting this value in (1) we get y p 1. This is the required differential equation. The order of this differential equation is 1 and degree is, degree is the power of highest power of differentiation. Eample 3 Form the differential equation of all circles of constant radius r h y k r...(1) In this equation there are arbitrary constant h and k. differentiating given equation w.r.t. we get h y k 0 () again differentiating (1) w.r.t. we get d y 1 y k 0..(3) Eliminating h and y k from equation (1), () and (3) we get r 3 3/ 1 1 or r d y d y This is the required differential equation of order and degree.
4 Solution of differential equation A solution of differential equation of differential equation is an equation which contains arbitrary constants as many as the order of differential equation and is called general solution. Other solutions obtained by giving particular values to the arbitrary constants in general solution are known as particular solutions. Methods to find the general solution of differential equation I. Variables separation II. Homogenous form III. Linear form Eample 4 Based on variables separation y dv put y v 1 dv 1 v dv c 1 v 1 tan v c or v tan( c) y tan( c) Eample 5 Solve y y Since in this equation sum of powers and y is and no constant term so this is the homogenous form For this put dv y v and v we get dv 1v dv 1v v or 1v 1v now applying variables are separable we get
5 1 v dv log 1 v v log log c 1 v c or v log 1 v log log c log y y c log y y / y c e This is the required solution of the given differential equation. Linear form of differential equation If the degree of the dependent variable and all derivatives is one, such differential equation is known as linear differential equation. 1. Differential equation of the form Py Q is known as first order linear equation where P and Q are functions of or constant. d y. Differential equation of the form P Qy R is known as second order linear differential equation and P, Q and R are functions of or constants. 3. Similarly differential equation of order n is defined. Solution of differential equation of first order is written as P P y e Q e C P Where e is known as Integrating factor. On multiplying both the side by I.F. we can easily find the solution of given differential equation. Integrating Factor of differential equation can be determined by different methods which we will discuss in net notes. Solution of higher order differential equation is given as y = Complementary function + Particular Integral
6 Eample 6 Solve y + log = 0 The given equation is written as y 1 log log 1.. I F e e 1 log Solution of D. E. y put log t y t then dt we get te dt Integrating by parts and simplifying y 1 log c which is required solution. For more problems for practice you can use your class XII book. Bernoulli s Equation Differential equation of the form n py Qy is known as Bernoulli s equation. For the solution of this equation divide both the sides by y n then dv put v n1 n y y 1 n dv we get (1 n) vp Q(1 n) Now this is linear form. Trajectories The family of curves in a plane is given by f(, y, c) = 0 depending on single parameter c. A curve making an fied angle ϕ at each of its point with a curve of family passing through that point is known as isogonal trajectory of other family and if θ = π/ then it is called orthogonal trajectory.
7 Eample 7 Find the orthogonal trajectory of y = 4a (a being parameter) First of all eliminate parameter from the given equation and its derivative the n replace by And then integrate you will find required orthogonal trajectory. Given y 4 a...(1) & y 4 a...() Now eliminating a from (1) & () we get y replacing by y 0 y C This is required orthogonal trajectory of the given trajectory. Now in Booklet we will discuss syllabus of Semester II
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