µ Differential Equations MAΘ National Convention 2017 For all questions, answer choice E) NOTA means that none of the above answers is correct.

Size: px

Start display at page:

Download "µ Differential Equations MAΘ National Convention 2017 For all questions, answer choice E) NOTA means that none of the above answers is correct."

Gyles Greene
5 years ago
Views:

1 µ Differential Equations MAΘ National Convention 07 For all questions, answer choice E) NOTA means that none of the above answers is correct.. C) Since f (, ) = +, (0) = and h =. Use Euler s method, we have 0 + ( ) ( h) = (0) + hf (0, (0)) = + =. C) Since + ( ) ( h) = ( h) + hf ( h, ( h)) = + = + ( ) 9 ( h) = ( h) + hf ( h, ( h)) = + = f (, ) = +, (0) = and h =. Use Euler s method, The end of this Euler strut is ( h, ( h)) = ( h, (0) + hf (0, (0))) = (, ) The first slope is the average of 5 f(0, ) + f(, ) + f (0, (0)) + f ( h, ( h)) 9 = = =. 8. C) It is eas to see implies c = 5 so the solution is t u. D) Let =, =, t h e n 5. E) u = sin t t cost + c which means, so the solution is ( ) ( ) = = = + c ( ) 5 = +. sin = can be rewritten as = sin cos +.. Condition ( ) = ( 5) = 0 = ( ) = 0,,. = sin cos + c. Condition ( ) 6 uu sin t =. The general solution is t u (0) = impliesc = =. The correct answer is D.

2 µ Differential Equations MAΘ National Convention A) The integrating factor of given equation is cos e. Multipl cos e and integrating the given cos cos equation, we have the general solution is e = c e. Condition ( ) = implies c = so cos cos the solution is ( + ) e =. It is eas to see ( + ) e = passes through point (,). The correct answer is A. d 7. D) The integrating factor is e = e. The correct answer is D 8. A) The integrating factor ise d implies c = so the solution is e d =. The general solution is = =. It is eas to see () 0 = c +. Condition (0, ) =. The correct answer is A sin 9. C) The integrating factor is. The general solution is 0 t = dt + c. Using 0 t sin t condition (,0) implies c = 0 dt = 0Si() so the solution is = 0(Si(0) Si()). The 0 t correct answer is C. 0. C) The general solution is sin = ( + ) + c. Using condition (0, ) implies c = so the solution is sin = ( + ). It is eas to see (0) =, ( ) =. 6. A) Since + = + = = ( ) = = + c Using condition (,) implies c = so the solution is ( ) =. The correct answer is A.. D) Rewrite the given equation in the standard form case is given b a + =. So the function f in this a + f (, ) =. The given equation is Euler homogeneous if and onl if c( a + ) f (, ) is scale invariant. It is eas to see f ( c, c) = = f (, ) for c( ) an real number a. The correct answer is D. ( + ). B) Rewrite the given equation in the form =.The general solution is ( + ) ( + ) = ln + c. Condition (, 0) implies c = so = ln +, () =. ( + ) ( + ) The correct answer is B.

3 µ Differential Equations MAΘ National Convention 07. B) It is eas to see N(, ) = +, M (, ) = a + bsin.the given equation is an eact if and M N onl if = a = =. Therefore, constant a =. B is correct answer. 5. B) It is eas to see (, ) N = e + cos, M (, ) = e sin.the equation is eact since M N = e sin =. e + cos = c is the general. B is correct answer. M N + 6. B) It is eas to see N(, ) = +, M (, ) = + and = = N ( + ) which is independent. The integrating factor is. B is correct answer. N M 7. C) Since = = which is M we multipl the given equation b independent. The integrating factor can be, the resulting equation is 5 d + ( + 0 ) d = 0. The general solution is answer. 6 0 c + =. C is correct 8. D) + = 0 can be rewritten in standard form + = 0. Use pd e = d, we have pd d e e = = =. D is correct answer. d d ln 9. E) The auiliar equation of given equation is r + r 5 = 0 has roots r = or r = 5. The 5 general solution is = Ae + Be. Initial conditions implies A=, B=, we have 5 = e + e. No correct answer was provided so the correct answer is E. 0. D) The auiliar equation of given equation is r + = 0 has roots r = i or r = i. The general solution is = Acos + Bsin. Initial condition onl implies B =, we have = Acos + sin (0) =. D is correct answer.. After. C) It is eas to see =satisfies equation and =, we have correct answer. sin c = +. Integrating + = 0 =. Integrating =, we have cos c = +. D is

4 µ Differential Equations MAΘ National Convention 07. C) The critical point is f (, ) = ( + )( )( ) = 0 =, =, =. C is correct answer.. D) From = +, (0) =, (0) =, we have (0) = and = + (0) = + ( ) =. D is correct answer.. A) From =, (0) =, () =, we have = + c b integrating. Substituting () = into general solution implies c =. That is = +. On the other hand, we have = =. Therefore, (0) = =. A is correct answer. (0) 5. D) From =, we have () = = ( ) = ( ) + = ( )( ) Therefore, () (0) = ( )( ) = 0. D is correct answer. 6. B) From given equation, we have = = Therefore, =, n = =, = 6 = 6, =!. B is correct answer. ( n) n + 7. B) The auiliar equation of given equation is form of particular solution is B is correct answer. p r r + = 0 has roots r r =. Substituting into given equation, we have A = =. We assume the p = e. 8. E) From the given equation = ( + ) +, we have 0 ( ) = + +, which implies = c ( + ) and + + = 0. The solution is = ( + ) c + c or = + A. Condition () = ( + ) implies A= 0, c=, The solution is = or =. = passes points (0,0) and ( + ) (, ) while = passes points (,0). E is correct answer.

5 µ Differential Equations MAΘ National Convention A) The auiliar equation is r + r = 0 has roots r = r = 0, r =. We assume the given equation has a particular solution in the form = ( Asin + Bcos ) e. Substituting into the given equation, we have p = ( sin + cos ) e. A is correct answer D) From given equation cot cos, ( c cos + = ) = 5, we have sin =. Condition ( 9 cos ) = 5 implies sin =, ( 5 cos ) = implies T sin =. ( ( ) ( )) =. The correct answer is D. T = 6 p

Basics Concepts and Ideas First Order Differential Equations. Dr. Omar R. Daoud

Basics Concepts and Ideas First Order Differential Equations Dr. Omar R. Daoud Differential Equations Man Phsical laws and relations appear mathematicall in the form of Differentia Equations The are one