Exercise Set 1

Exercise Set 1 - Exercise Set 1 Math 4027 Due February 1...

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Exercise Set 1 Math 4027 Due: February 1, 2005 1. Find the solution of the initial value problems: (a) y 0 + 2 y = x, y (0) = 1, I Solution. Multiply by e 2 x to get ( e 2 x y ) 0 = xe 2 x and then integrate to get e 2 x y = x 2 e 2 x - 1 4 e 2 x + C. Solve for y and substitute y (0) = 1 to get C = 5 / 4. Hence y = 1 4 ( 2 x - 1 + 5 e - 2 x ) . J (b) y 0 - y = sin x, y (0) = 2. I Solution. Multiply by e - x to get ( e - x y ) 0 = e - x sin x and then integrate to get e - x y = - 1 2 e - x (sin x + cos x ) + C. Solve for y and substitute y (0) = 2 to get C = 5 / 2. Hence y = 5 2 e x - 1 2 (sin x + cos x ) . J 2. Find the general solution of (a) y 0 + 2 y = x 2 , I Solution. Multiply by e 2 x to get ( e 2 x y ) 0 = x 2 e 2 x and then integrate to get e 2 x y = x 2 2 e 2 x - x 2 e 2 x + 1 4 e 2 x + C. Solve for y to get y = 1 4 ( 2 x 2 - 2 x + 1 ) + Ce - 2 x . J (b) y 0 - y = sin 2 x . I Solution. Multiply by e - x to get ( e - x y ) 0 = e - x sin 2 x and then integrate to get e - x y = - 1 5 e - x (sin 2 x + 2 cos 2 x ) + C. Solve for y to get y = Ce x - 1 5 (sin 2 x + 2 cos 2 x ) . J 1
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Exercise Set 1 Math 4027 Due: February 1, 2005 3. Find the general solution of y 0 + ay = e bx , where a and b
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