Differential Equation/Examples/Second Order Second Degree Non-Linear Ordinary

Example of Differential Equation

Ordinary non-linear differential equation of the $2$nd order and $2$nd degree:

$\paren {1 + \paren {\dfrac {\d y} {\d x} }^2}^{3/2} = 3 \dfrac {\d^2 y} {\d x^2}$

Proof

Squaring both sides gives:

$\paren {1 + \paren {\dfrac {\d y} {\d x} }^2}^3 = 9 \paren {\dfrac {\d^2 y} {\d x^2} }^2$

It is seen that the term in $\dfrac {\d^2 y} {\d x^2}$ is squared.

Hence the result by definition of degree.

$\blacksquare$

Sources

• 1926: E.L. Ince: Ordinary Differential Equations ... (previous) ... (next): Chapter $\text I$: Introductory: $\S 1.1$ Definitions
• 1952: H.T.H. Piaggio: An Elementary Treatise on Differential Equations and their Applications (Revised ed.) ... (previous) ... (next): Chapter $\text I$: Introduction and Definitions. Elimination. Graphical Representation: $1 \ (3)$