3.5 Euler ode \(x^{2}y^{\prime \prime }+xy^{\prime }+y=f\left ( x\right ) \)

ode internal name "second order euler ode"

Solved by substitution \(y=x^{r}\) and solving for \(r\). Solution will be \(y=c_{1}x^{r_{1}}+c_{2}x^{r_{2}}\) where \(r_{1},r_{2}\) are the roots of the characteristic equation. For repeated root, the second solution is multiplied by extra \(\ln \left ( x\right ) \) and not extra \(x\) as is the case with standard constant coefficient ode. The particular solution is found in the same way using variation of parameters. Can not use undetermined coefficient method since this is not constant coefficients ode. The basis functions here are \(x^{r_{1}},x^{r_{2}}\) if not repeated roots, else the basis are \(x^{r_{1}},\ln \left ( x\right ) x^{r_{2}}\).

Initial conditions for Euler ode can not be at \(x=0\).  For ode of the form

\[ \left ( x-a\right ) ^{2}y^{\prime \prime }+\left ( x-a\right ) y^{\prime }+y=f\left ( x\right ) \]

This is still Euler ode. We start by substitution \(X=x-a\) which gives

\[ X^{2}y^{\prime \prime }+Xy^{\prime }+y=f\left ( X+a\right ) \]

This is now solved using \(y=X^{r}\) as before. When we obtain the solution \(y\left ( X\right ) \) then every \(X\) is replaced back by \(x-a\) to obtain the final solution. Below are two examples.