Solve differential equation with variation of parameters

Question

I'm having a little trouble understanding how to solve the following differential equation. The equation that has to be solved is: $$y'' - y' - 2y = 2e^{-t}$$

The problem I am having is that I don't understand why they equate that part with the derivatives of the $u$ parameters to $0$. See the problem below

Example 13. Problem 2: $y''-y'-2y=2e^{-t}$. The characteristic equation is: $$r^2-r-2=0\iff (r-2)(r-1)=0\implies y_h(t)=c_1e^{2t}+c2e^{-t}.$$ Suppose that $y(t)=u_1(t)e^{2t}+u_2(t)e^{-t}$, then it follows: $$ y'(t)=\underbrace{u_1'(t)e^{2t}+u_2'(t)e^{-t}}_{=0}+2u_1(t)e^{2t}-u_2(t)e^{-t} $$

Here they first find the characteristic equation and write down the general solution. They then replace the constants with the parameter "$u$" and take the derivative.

As you can see, they just say that the derivative part of the $u$ parameter is equal to $0$. But why? How? Where did that come from? I can't find it anywhere in my book.

It's probably a facepalm answer but I would really appreciate it.

Answer 1

Read through the derivation of the method of variation of parameters here and it should clarify things. The very method is predicated on finding $u_1(x)$, $u_2(x)$ such that $y_p(x)=u_1(x)y_1(x)+u_2(x)y_2(x)$ is a solution of the nonhomogeneous problem, where $y_1(x)$, $y_2(x)$ are known (linearly independent) solutions of the homogeneous problem.

So we are looking for the yet-to-be-determined functions $u_1$ and $u_2$. While doing so, we can employ any constraints we like; such constraint(s) will be fruitful so long as they ultimately accomplish our goal of finding $u_1$, $u_2$ for which $y_p$ above is a solution to the nonhomogeneous problem.

Now your question may really be, "How did they think to do this? Because I wouldn't have thought to do that." Fair enough question/comment. However, now that you have seen this "trick" and that it "worked" in the sense of eventually finding you a second solution, just rejoice and use the trick again (while hopefully understanding that it was mathematically legitimate to impose that constraint to make the second derivative simpler).