Change of Variables

Description

Change of variables describes a tool used to transform a Ordinary differential equation into a better known differential equation.

Definition (sketch)

Let $x^{\prime }=f(t,x)$ be a order-one ODE. We define a new variable $u(t,x)$ as some function in $t$ and $x$. Calculating $u^{\prime }$ using the chain rule and rewriting $t,x$ in terms of $u$ is called a variable transformation. There are some rules. I think $u$ needs to be at least a $C^1$-Diffeo in order to give reversible properties

Most likely, its best to look at examples.

Properties

Tip

Examples

First-order Differential equation equivariant under scaling

Let $x^{\prime }=f(t,x)$ be a differential equation fulfilling $f(t,x)=f(\sigma t,\sigma x)$ for all $0\ne \sigma \in \mathbb{R}$. Then the substitution $y=\frac{x}{t}$ results in the differential equation: $y^{\prime }=\frac{f(1,y)}{t}$ Which can be solved by Separation of variables.

Proof: We calculate $y^{\prime }=\frac{t{x}^{\prime }-x}{t^2}=\frac{tf(t,x)}{t^2-x}=\frac{f(\frac{t}{t},\frac{x}{t})-\frac{x}{t}}{t}=\frac{f(1,y)-y}{t}$

Affine linear transformation

A differential equation of the form $x^{\prime }=g(\alpha t+\beta x+\gamma )$ can be transformed using the substitution $y=\alpha t+\beta x+\gamma$ into the Linear ODE $y^{\prime }=\alpha +\beta g(y)$

Some radially symmetric problems are much easier to solve in Polar coordinates.

Polar coordinates

We can make use of the variable transform $(x,y)=(r\cos (\theta ),r\sin (\theta ))$ to simplify a problem. The transformation has the determinant $r$ and the backtransformation is given by $(r,\theta )=(\Vert (x,y)\Vert ,\arctan \frac{y}{x})$