# Solution to Differential Equation/Examples/Arbitrary Order 2 Degree 3 ODE

## Examples of Solutions to Differential Equations

Consider the equation:

$(1): \quad y = x^2$

where $x \in \R$.

Then $(1)$ is a solution to the second order ODE:

$(2): \quad \paren {y''}^3 + \paren {y'}^2 - y - 3 x^2 - 8 = 0$

defined on the domain $x \in \R$.

## Proof

 $\ds y$ $=$ $\ds x^2$ $\ds \leadsto \ \$ $\ds y'$ $=$ $\ds 2 x$ Power Rule for Derivatives $\ds \leadsto \ \$ $\ds y''$ $=$ $\ds 2$ Power Rule for Derivatives

Then:

 $\ds$  $\ds \paren {2}^3 + \paren {2 x}^2 - x^2 - 3 x^2 - 8$ substituting for $y$, $y'$ and $y''$ from above into the left hand side of $(1)$ $\ds$ $=$ $\ds 8 + 4 x^2 - x^2 - 3 x^2 - 8$ $\ds$ $=$ $\ds 0$ which equals the right hand side of $(1)$

$\blacksquare$