Solution to Differential Equation/Examples/Arbitrary Order 1 ODE: 3

Examples of Solutions to Differential Equations

Consider the equation:

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

for all $x \in \R$.

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

$y' = x$

Proof

We have that:

\(\text {(1)}: \quad\)

\(\ds y\)

\(=\)

\(\ds \dfrac {x^2} 2\)

by hypothesis

\(\ds \leadsto \ \ \)

\(\ds y'\)

\(=\)

\(\ds \dfrac {2 x} 2\)

Power Rule for Derivatives

\(\ds \)

\(=\)

\(\ds x\)

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): differential equation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): differential equation