3(c)

Using (a) and (b), the general solution of the differential equation is $$y = {\rm e}^{-x} (\cos x - \sin x)/8 + c_1 {\rm e}^x \cos x + c_2 {\rm e}^x \sin x$$. Then $$y' = - {\rm e}^{-x} \cos(x)/4 + (c_1 + c_2) {\rm e}^x \cos x + (-c_1 + c_2) {\rm e}^x \sin x$$. From the initial conditions we have $y(0) = 1/8 + c_1 = 0$ and $y'(0) = c_1 + c_2 - 1/4 = 1$, so $c_1 = -1/8$ and $c_2 = 11/8$. So the solution is

$$y = \frac{(\cos x - \sin x) {\rm e}^{-x} + (-\cos x + 11 \sin x) {\rm e}^x}{8}$$