Differential Equations Math 285 Hour Exam 2 Fall, 2011

1.	a. Are the three functions y1(x) = 1, y2(x) = sin^2x, y3(x) = 2 cos^2 x linearly dependent? b. Is it possible that the above functions satisfy an equation y” + p(x) y’ + q(x) y = 0 on the entire line, where the functions p(x) and q(x) are continuous on the entire line? Answers: a. linearly dependent b. No
2.	Consider a differential equation any^n + . . . . + aoy = f(x). Let the characteristic equation have the roots 0, 0, - 3, 1 ± 2i, 1 ± 2i, 1 ± 2i a. Give a general solution of the corresponding homogeneous equation. b. Give a particular solution yp(x) with undetermined coefficients if f(x) = e ^-3x cos 2x + 5 x e ^-3x sin 2x c. Give a particular solution yp(x) with undetermined coefficients if f(x) = x e^x cos 2x - 5 e^xsin 2x Answers: a. y = C1 + C2 x + C3 e ^-3x + e ^x (C4 + C5 x + C6 x^2) cos 2x + e ^x (C7 + C8 x + C9 x^2) sin 2x b. yp(x) = e ^-3x (A + B x) cos 2x + e ^-3x (C + D x) sin 2x c. yp(x) = x^3 [e ^x (A + B x) cos 2x + e ^x (C + D x) sin 2x ]
3.	a. Find a particular solution to the equation y “ + 9 y = ( 6x – 7 ) e ^3x b. Find the solution to the initial value problem for the above equation With initial conditions y(0) = 0, y ‘(0) = 1/3 Answers: a. yp = [ (1/3) x – ½ ] e ^3x and b. y = [ (1/3) x – ½ ] e ^3x + ½ cos 3x + ½ sin 3x Click here to continue with this exam.

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