# Differential Equation/Examples/Fourth Order First Degree Non-Linear Ordinary

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## Example of Differential Equation

Ordinary non-linear differential equation of the $4$th order and $1$st degree:

- $x \dfrac {\d^4 y} {\d x^4} + 2 \dfrac {\d^2 y} {\d x^2} + \paren {x \dfrac {\d y} {\d x} }^5 = x^3$

## Proof

The term in $\dfrac {\d y} {\d x}$ is raised to the $5$th degree and so the equation is non-linear.

The term in $\dfrac {\d^4 y} {\d x^4}$ is raised to the $1$st degree.

The result follows by definition of degree.

$\blacksquare$

## Sources

- 1963: Morris Tenenbaum and Harry Pollard:
*Ordinary Differential Equations*... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $3$: The Differential Equation: $(3.18)$