Lecture 2

Singular Solutions

Example 1

$$x{\left(\frac{dy}{dx}\right)}^2-3y\left(\frac{dy}{dx}\right)+q{x}^2=0$$

Example 2

$${\left(\frac{dy}{dx}\right)}^3-4xy\left(\frac{dy}{dx}\right)+8y^2=0$$

Summary

We’ve done Singular Solutions and Family Solutions

Given a first-order second-degree ODE of the form

$$F\left(x,y,\frac{dy}{dx}\right)=0$$

Where we solved some special cases. In general, it is not solvable.

The way we solve using special cases is to eliminate either $x$ or $y$

$$\Rightarrow \{\begin{array}{l}y=g(x,p)\to p=\frac{dy}{dx}=\frac{\partial g}{\partial x}+\frac{\partial g}{\partial p}\cdot \frac{\partial p}{\partial x}\\ x=g(y,p)\to \frac{1}{p}=\frac{dx}{dy}=\frac{\partial g}{\partial y}+\frac{\partial g}{\partial p}\cdot \frac{\partial p}{\partial y}\end{array}\text{where }p=\frac{dy}{dx}$$

Next, we will look to factorize. After that, you will get your singular solution and family of solutions