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 $(1)$

$\blacksquare$

Sources

• 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $3$: The Differential Equation: Example $3.5$