Calc 2 General Solution using Method of Undetermined Coefficients?

Question

Hiya!

The question is:
Solve the differential equation using the method of undetermined coefficients.
y'' + 2y' + y = xe-x

The auxiliary equation is:
r^2 + 2r + 1

The complementary solution is:
c1xe^-x + c2e^-x

But what is the general solution? I tried:
y = Axe^-x
Take the derivative:
y' = -Ae^-x(x - 1)
Take the second derivative:
y" = Ae^-x (x - 2)

Plug it back into the original equation:
1 * Ae^-x (x - 2) + 2 * -Ae^-x(x - 1) + 1 * Axe^-x = xe^-x
Axe^-x - 2Ae^-x - 2Axe^-x + 2Ae^-x + Axe^-x - xe^-x = 0
All of the things with A cancel, leaving just -xe^-x = 0...

What did I do wrong?

Thanks a lot!

? · Accepted Answer

You need to use the general solution of the form y = (Ax + B)*(e^(-x))

That's because the lone term x is actually a polynomial and must be expanded to represent that. For example x^2 is actually representative of the larger form a*x^2 + b*x + c where a = 1, b = 0 and c = 0. If you expand that to this case you see that e^(-x) is actually a*e^(-x), x is actually (a*x + b). Combine and collapse coefficients.

If you don't understand the choice of the general solution I suggest you take a look at solving using method of undetermined coefficients in your differential equations book. I suggest Elementary Differential Equations and Boundary Value Problems by Boyce/DiPrima.

You need to look at the homogeneous equation solution. In this case you see that the term x*e^(-x) appears in the homogeneous equation and in your proposed solution. (Formally, a proposed solution must be of the form x^(n)*f(x) for all n such that n is one degree larger than the largest polynomial degree of the homogeneous solution.) Therefore you need to get rid of that similarity by multiplying by x^2.

So you have x^2*(Ax + B)(e^-x). 
Differentiate and solve. You should get (1/6)*x^3*(e^(-x))

Calc 2 General Solution using Method of Undetermined Coefficients?

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