Question 1
Use the method of variation of parameters to find the general solution of the nonhomogeneous linear equation x
2y" - (2x + x
2)y' + (2 + x)y = 2x
3e
-x given that y
1(x) = x and y
2(x) = xe
x are independent solutions of the corresponding homogeneous equation.
Question 2
Find the general solution of the differential equation y"+ y = csc x.
◦ C
1 cos x + C
2 sin x + x cos x + sin x ln (sin x)
◦ C
1 cos x + C
2 sin x + x sin x + cos x ln (cos x)
◦ C
1 cos x + C
2 sin x - x cos x + sin x ln (sin x)
◦ C
1 cos x + C
2 sin x - x cos x - sin x ln (sin x)
◦ C
1 cos x + C
2 sin x - x sin x + cos x ln (cos x)