Toronto Math Forum
MAT2442014F => MAT244 MathTests => TT1 => Topic started by: Victor Ivrii on October 09, 2014, 02:01:12 AM

Find a particular solution of
\begin{equation*}
x^2 y''(x)  6y(x)=10x^{2}  6, \qquad x >0 .
\end{equation*}

Observe that the equation is an Euler equation, then we let
\begin{equation*} t = \log x
\end{equation*}
Then the equation becomes
\begin{equation*}
y''  y'  6y =10 e^{2t} 6
\end{equation*}
It is the same ODE from question 3 (http://forum.math.toronto.edu/index.php?topic=445.0).
Use the particular solution from 3 i.e
\begin{equation*}
y=2te^{2t} + 1
\end{equation*}
Plug in $x$ back, we have
\begin{equation*} y=\frac{2\log x}{x^2} + 1. \end{equation*}

I made some minor editing. Note that \log x, \sin ,\cos รข. to keep them upright and provide a proper spacing.
But what would be a general solution? (I know it was not required in the test). Also I would like to see on the forum solution without substitution: i.e. we are looking at $y_p=y_{p1}+y_{p2}$, $y_{p1}=Ax^{2}\ln x$ (because $r=2$ is a characteristic root) and $y_{p2}=B$.