Separable ODEs

Definition: Separable Ordinary Differential Equation

An $n$-th order ordinary differential equation is separable if it can be expressed in the following form:

$$N\left(y^{(n-1)}(x)\right)\cdot y^{(n)}(x)=M(x)$$

First-Order Separable ODEs

A first-order separable ODE can be expressed in the following form:

$$N(y(x))\cdot y^{\prime }(x)=M(x)$$

Algorithm: Solving First-Order Separable ODEs

We are given a first-order separable ODE:

$$N(y(x))\cdot y^{\prime }(x)=M(x)$$

1. Integrate both sides with respect to $x$:

$$\int N(y(x))\cdot y^{\prime }(x)\mathrm{dx}=\int M(x)\mathrm{dx}$$

2. Apply integration by substitution on the left-hand side:

$$\int N(u)\mathrm{du}=\int M(x)\mathrm{dx}$$

1. If $\int N(u)\mathrm{du}$ and $\int M(x)\mathrm{dx}$ have closed form expressions, substitute $y(x)$ back into $u$ after integrating to obtain an expression for $y(x)$. Otherwise, the solutions remains in the above implicit form and must be found numerically.

EXAMPLE

Consider the following equation:

TODO