Question

Which of the following functions are solutions of the differential equation y" + y = 7 sin(x)? (Select all that apply.)

Answer 1

The homogeneous equation

y'' + y = 0

has characteristic equation

r² + 1 = 0

with roots at r = ± √(-1) = ± i, and so the characteristic solution is

y = C₁ cos(x) + C₂ sin(x)

Using the method of undetermined coefficients, suppose

y = a x cos(x) + b x sin(x)

(we would normally consider y = a cos(x) + b sin(x), but cos(x) and sin(x) are already part of the characteristic solution)

The derivatives are

y' = a cos(x) - a x sin(x) + b sin(x) + b x cos(x)

y' = (b - a x) sin(x) + (a + b x) cos(x)

and

y'' = -a sin(x) + (b - a x) cos(x) + b cos(x) - (a + b x) sin(x)

y'' = (-2a - b x) sin(x) + (2b - a x) cos(x)

Then

y'' + y = (-2a - b x) sin(x) + (2b - a x) cos(x) + a x cos(x) + b x sin(x)

7 sin(x) = -2a sin(x) + 2b cos(x)

and it follows that a = -7/2 and b = 0.

So, the general solution to the DE is

y = C₁ cos(x) + C₂ sin(x) - 7/2 x cos(x)

which means the only valid solutions from the given choices are

y = 7 sin(x)

y = 7 cos(x)