Hello everyone(adsbygoogle = window.adsbygoogle || []).push({});

The following two differential equations appear on page 93 of George Simmons' book on Differential Equations. While I have been able to solve them, I have some questions. [Not HW]

I. [tex](x^2-1)y'' - 2xy' + 2y = (x^2-1)^2[/tex]

II. [tex](x^2 + x)y'' + (2-x^2)y' - (2+x)y = x(x+1)^2[/tex]

How I solved them:

For the first equation, [itex]y = x[/itex] is a solution of the homogeneous part as can be verified by inspection. From this we can construct a second solution [itex]y = xv(x)[/itex] with [itex]v(x)[/itex] given by

[tex]v(x) = \int \frac{1}{x^2}e^{\int \frac{2x}{x^2-1}dx}dx = \left(x + \frac{1}{3x^3}\right)[/tex]

The homogeneous equation thus has the solution

[tex]y_{h}(x) = c_{1}x + c_{2}\left(x^2 + \frac{1}{3x^2}\right)[/tex]

For the general solution, add a polynomial and compare coefficients.

For the second equation, [itex]y = 1/x[/itex] is a solution of the homogenous part. The second solution is [itex]y = v(x)/x[/itex] where [itex]v(x)[/itex] is given by

[tex]v(x) = \int x^2 e^{-\int \frac{2-x^2}{x^2 + x} dx}dx = \int x^2 \left(\frac{1+x}{x^2}\right) e^{x}dx = \int (x+1)e^{x}dx = xe^{x}[/tex]

The general solution to the homogenous part is thus

[tex]y_{h}(x) = \frac{c_{1}}{x} + c_{2}e^{x}[/tex]

The general solution for the nonhomogeneous equation can be constructed as for the first part.

Now, I have two questions:

1. I could "guess" the simple solutions to the homogeneous equations in both cases, but how do I get to this solution rigorously? (I tried some substitutions but none of them worked.)

2. Is there some totally different way out to solve these equations too?

Thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Alternate Method for solving two Second Order DEs.

**Physics Forums | Science Articles, Homework Help, Discussion**