Find solution to second order linear differential equation in 3 parts.

Question

Say I have this equation:

$$y'' - 4y' + 4y = x - \sin{x}$$

My process is to: - find complementary solution - find the particular solution in two parts - add them together to find general solution.

Is this process and answer correct?

complementary solution find auxiliary: $$r^2 - 4r + 4 = 0$$ $$(r-2)(r-2) = 0$$

so complementary is: $y_c = c_1e^{2x} + c_2xe^{2x}$

part 1 of particular:

$$y_p1 = y'' - 4y' + 4y = x$$

the particular is in the form: $Ax + B$ $$y'p = A$$ $$y''p = 0$$ so via substitution:

$$- 4A + 4Ax + 4B = x$$

setting coefficients equal:

$-4A = 1$ and $A = \frac{1}{4}$ and $B = \frac{1}{4}$

so this part of this particular solution is $$y_p1 = \frac{1}{4}x + \frac{1}{4}$$

the other part of the particular is this:

$$y_p2 = A\sin{x} + B\cos{x}$$ $$y'_p2 = A\cos{x} - B\sin{x}$$ $$y''_p2 = -A\sin{x} - B\cos{x}$$

so substituting into $y'' - 4y' + 4y = x$:

$$-A\sin{x} - B\cos{x} - 4A\cos{x} + 4B\sin{x} + 4A\sin{x} + 4B\cos{x} = -\sin{x}$$

so for the sines: $$3A + 4B = -1$$ for the cosines: $$-3B - 4A = 0$$ solving: $$-3B = 4A$$ $$\frac{-3}{4}B = A$$ $$\frac{-9}{4}B + 4B = -1$$ $$\frac{7}{4}B = -1$$ $$B = \frac{-4}{7}$$ $$A = \frac{12}{28}$$

so summing all together:

general solution

$$y = c_1e^{2x} + c_2xe^{2x} + \frac{1}{4}x + \frac{1}{4} + \frac{12}{28}\sin x -\frac{4}{7}\cos x$$

Is this right?

Answer 1

You solved for the coefficients $A,B$ wrong in the second $y_p$. You should have the system

$$3A+4B = -1 \;\;\;\;\; -4A + 3B = 0$$

You instead have $-3B$. The correct coefficients would then be

$$A = -\frac{3}{25} \;\;\;\;\; B = - \frac{4}{25}$$

Aside from this, the overarching idea and the rest of your work seems to be correct, as per usual.

If I did have to nitpick one thing, you shouldn't have $y_p1 = ...$ and $y_p2 = ...$, setting them equal to the ODE. That could cause both you and your reader confusion. Maybe say instead say "$y_p^{(1)} = y$ is the solution to this ODE". (The different placement for the $(1)$ index is arguably more common, or, at least, it's how I've seen it written when necessary.)