Cauchy-Euler Equation

A linear equation of the form

$$a_n{t}^n{y}^{(n)}+a_{n-1}{t}^{n-1}{y}^{(n-1)}+\cdots +a_{1}t{y}^{\prime }+a_{0}y=f(t)$$

where the $a$ values are constants

Solving Homogeneous

1. Find the characteristic/auxiliary/identity equation
   • $a{t}^2{y}^{\prime \prime }+bt{y}^{\prime }+cy=0\to a{r}^2+(b-a)r+c=0$
2. Solve for roots
3. Turn the roots into solutions
   • Distinct Real Roots: $y_1=t^{r_1},y_2=t^{r_2}$
   • One Repeated Root: $y_1=t^r,y_2=t^r\ln t$
     • $\ln t$ is a result of the Reduction of Order Thm
   • Complex Roots
     • $r=\alpha \pm i\beta$
     • $y_1=t^{\alpha }\cos (\beta \ln t),y_2=t^{\alpha }\sin (\beta \ln t)$
4. $y_h$ is linear combination of those

Solving Non Homogeneous

1. Find $y_1$, $y_2$
2. Find particular solution using Variation of Parameters