01 - ODEs

ODE

Equation for an unknown function $y(x)$ of one variable $x$ in form

$$G(x,y(x),y^{\prime }(x),y^{\prime \prime }(x),\cdots ,y^{(n)}(x))=0$$

where $y^{(n)}(x)=\frac{d^{n}y}{d{x}^n}(x)$

This can be solved for the highest derivative of $t$ in the form

$$y^{(n)}(x)=F(x,y,y^{\prime },\cdots ,y^{(n-1})$$

Order of an ODE

Order $n$ of an ODE is order of the highest derivative that appears

Non Uniqueness of solutions to ODEs with initial values

Consider IVP

$$y^{\prime }(x)=3y(x{)}^{2/3},y(0)=0$$

By separation of variables you get

$$\int \frac{dy}{3y^{2/3}}=\int dx$$

Solving that you get

$$y(x)=(x+A{)}^3\text{ where }A\text{ is a constant}$$

Using initial condition $y(0)=0$ we get

1. Solution $y(x)=x^3$
2. Trivial Solution $y(x)=0$
3. Infinite Solutions for $c,d$ where $c\le 0\le d$ then

$$y_{c,d}(x):=\{\begin{array}{ll}(x-c{)}^3& \text{ for }x<c\\ 0& \text{ for }c\le x\le d\\ (x-d{)}^3& \text{ for }x>d\end{array}$$

All these functions satisfy the initial conditions and are differentiable everywhere