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

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: $(3.18)$