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MATH 3321 Sample Questions for Exam 2 Linear Nonhomogeneous Diﬀerential Equations 1. Find the general solution of y ±± - 4 x y ± + 6 x 2 y = 4 x 2 . Answer y = C 1 x 2 + C 2 x 3 + 2 3 . 2. Find the general solution of y ±± +4 y =2tan2 x . Answer y = C 1 cos 2 x + C 2 sin 2 x - 1 2 sin 2 x ln | sec 2 x + tan 2 x | . 3. Find the general solution of y ±± - 6 y ± +9 y =4 e 3 x + e 3 x x . Answer y = C 1 e 3 x + C 2 xe 3 x +2 x 2 e 3 x + 3 x ln x . 4. Find the general solution of y ±± y =4cos2 x . Answer y = C 1 cos 3 x + C 2 sin 3 x + 4 5 cos 2 x . 5. Find the general solution of y ±± y =2s in2 x . Answer y = C 1 cos 2 x + C 2 sin 2 x - 1 2 x cos 2 x . 6. Find the general solution of y ±± - 6 y ± +8 y =2 e 4 x +6. Answer y = C 1 e 4 x + C 2 e 2 x 4 x + 3 4 . 7. A particular solution of the nonhomogeneous diﬀerential equation y ±± - 2 y ± - 15 y =2cos3 x +5 e 5 x will have the form: Answer z = A cos 3 x + B sin 3 x + Cxe 5 x + D . 8. A particular solution of the nonhomogeneous diﬀerential equation y ±± - 8 y ± +16 y = e 2 x sin 4 x e 4 x x will have the form: Answer Ae 2 x cos 4 x + Be 2 x sin 4 x + Cx 2 e 4 x + Dx + E . 1

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Laplace Transformations 1. Find the Laplace transform of f ( x )=2 e - 3 x + cos 2 x +5 x .
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