Table of Contents
Differential Equations
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Topic PlaylistA Cauchy-Euler equation is a linear homogeneous ordinary differential equation with every $y^{(n)}(x)$ derivative of the unknown function multiplied by a constant and the independent variable $x^{n}$. Similarly to Bernoulli equations, Cauchy-Euler equations are a rare case of a solvable complicated ODE that the equation has been named. One of their uses is solving Laplace's equation in polar coordinates.
Explanation VideoA Cauchy-Euler equation is an ODE of the form,
\begin{equation} a_{n}x^{n}y^{(n)}(x)+a_{n-1}x^{n-1}y^{(n-1)}(x)+ \cdots +a_{0}y(x) = 0 \end{equation}One way to solve them is guessing a homogenous solution of the form $y(x)=x^{r}$ and determining $r$, similar to guessing $y(x)=e^{rx}$ for equations of the form $ay''+by'+cy=0$. Another method is a clever change of variables.
The solutions are often defined only for $x>0$, but can be extended to everything except $x=0$ by replacing $x$ with $|x|$.
Explanation VideoSimilar to constant coefficient, there are three cases when solving the polynomial for $r$ after substituting $y(x) = x^{r}$ into the ODE.
Distinct Real Roots $r_{1}$ and $r_{2}$
\begin{equation} y_{1}(x) = x^{r_{1}} \qquad y_{2}(x) = x^{r_{2}} \end{equation}Repeated Real Root $r$
\begin{equation} y_{1}(x) = x^{r} \qquad y_{2}(x) = \ln(x)x^{r} \end{equation}Imaginary Roots $\alpha \pm \beta i$
\begin{equation} y_{1}(x) = x^{\alpha}\cos\big(\beta\ln(x)\big) \qquad y_{2}(x) = x^{\alpha}\sin\big(\beta\ln(x)\big) \end{equation} Solution VideoFind the general solution to the ODE by determining $r$ for the guess $y(x)=x^{r}$.
\begin{equation} x^{2}y'' - 3xy' + 3y = 0 \end{equation} Solution VideoFind the general solution to the ODE.
\begin{equation} x^{2}y'' + 3xy' + y = 0 \end{equation} Solution VideoFind the general solution to the ODE.
\begin{equation} x^{2}y'' + 3xy' + 2y = 0 \end{equation}