MIT18_03S10_1ex

# MIT18_03S10_1ex - 18.03 EXERCISES 1. First-order ODEs 1A....

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18.03 EXERCISES 1. First-order ODE’s 1A. Introduction; Separation of Variables 1A-1. Verify that each of the following ODE’s has the indicated solutions ( c i , a are con- stants): a) y 2 y + y = 0 , y = c 1 e x + c 2 xe x sin x + a b) xy + y = x sin x, y = cos x x 1A-2. On how many arbitrary constants (also called parameters ) does each of the following families of functions depend? (There can be less than meets the eye . . . ; a, b, c, d, k are constants.) a) c 1 e kx b) c 1 e x + a c) c 1 + c 2 cos 2 x + c 3 cos 2 x d) ln( ax + b ) + ln( cx + d ) 1A-3. Write down an explicit solution (involving a deFnite integral) to the following initial-value problems (IVP’s): x 1 ye a) y = y 2 ln x , y (2) = 0 b) y = , y (1) = 1 x 1A-4. Solve the IVP’s (initial-value problems): a) y = xy + x , y (2) = 0 b) du = sin t cos 2 u, u (0) = 0 y dt 1A-5. ±ind the general solution by separation of variables: dv a) ( y 2 2 y ) dx + x 2 dy = 0 b) x = 1 v 2 dx ± ² 2 dx c) y = y 1 d) = 1 + x x + 1 dt t 2 + 4 1B. Standard First-order Methods 1B-1. Test the following ODE’s for exactness, and Fnd the general solution for those which are exact. a) 3 x 2 y dx + ( x 3 + y 3 ) dy = 0 b) ( x 2 y 2 ) dx + ( y 2 x 2 ) dy = 0 c) ve uv du + ye uv dv = 0 d) 2 xy dx x 2 dy = 0 1B-2. ±ind an integrating factor and solve: 2 x a) 2 x dx + dy = 0 b) y dx ( x + y ) dy = 0 , y (1) = 1 y c) ( t 2 + 4) dt + t dx = x dt d) u ( du dv ) + v ( du + dv ) = 0 . v (0) = 1 1

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2 18.03 EXERCISES 1B-3. Solve the homogeneous equations dw 2 uw a) y = 2 y x b) = y + 4 x du u 2 w 2 c) xy dy y 2 dx = x x 2 y 2 dx y 4 + xy 2 1B-4. Show that a change of variable of the form u = turns y = into an x n x 2 y equation whose variables are separable, and solve it. (Hint: as for homogeneous equations, since you want to get rid of y and y , begin by expressing them in terms of u and x .) 1B-5. Solve each of the following, Fnding the general solution, or the solution satisfying the given initial condition. dx t a) xy + 2 y = x b) dt x tan t = , x (0) = 0 cos t c) ( x 2 1) y = 1 2 xy d) 3 v dt = t ( dt dv ) , v (1) = 1 4 dx 1B-6. Consider the ODE + ax = r ( t ), where a is a positive constant, and lim r ( t ) = 0 . dt t ±² Show that if x ( t ) is any solution, then lim x ( t ) = 0 . (Hint: use L’Hospital’s rule.) t ±² 1B-7. Solve y = y . Hint: consider dx . y 3 + x dy n 1B-8. The Bernouilli equation. This is an ODE of the form y + p ( x ) y = q ( x ) y , n = 1 . Show it becomes linear if one makes the change of dependent variable u = y 1 n . (Hint: begin by dividing both sides of the ODE by y n .) 1B-9. Solve these Bernouilli equations using the method decribed in 1B-8: 3 a) y + y = 2 xy 2 b) x 2 y y = xy 1B-10. The Riccati equation. After the linear equation y = A ( x ) + B ( x ) y , in a sense the next simplest equation is the Riccati equation y = A ( x ) + B ( x ) y + C ( x ) y 2 , where the right-hand side is now a quadratic
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## This note was uploaded on 10/21/2011 for the course MATH 18.03 taught by Professor Vogan during the Fall '09 term at MIT.

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MIT18_03S10_1ex - 18.03 EXERCISES 1. First-order ODEs 1A....

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