Answer to Question #50894 in Differential Equations for Nayan Jyoti Das

Question

3. a) Write a suitable form of particular solution for solving the equation
y'' + 3y' + 2y = e^x (x^2 + 1) sin 2x + 3e^-x cos x + 4e^x
by the method of undetermined coefficients.

b) Verify that e^x and xe^x are solutions of the homogeneous equation corresponding to the
equation
y''− 2y'+ y = e^x / (1 + x^2), −&<x<&
and find the general solution using the method of variation of parameters.

c) If y1 = 2x+2 and y2 = -x^2 / 2 are the solutions of
xy' + y' - y'^2 / 2
then are the constant multiples c1 y1 and c2 y2 , where c1 and c2 are arbitrary, also solutions
of the given differential equation? Is the sum y1 + y2 a solution? Justify your answer.