Question

For the following differential equation verify that the accompanying function is a solution:

Differential equation	Function
\[y = \left( \frac{dy}{dx} \right)^2\]	\[y = \frac{1}{4} \left( x \pm a \right)^2\]

Answer 1

We have,

\[y = \frac{1}{4} \left( x \pm a \right)^2 . . . . . \left( 1 \right)\]

Differentiating both sides of (1) with respect to x, we get

\[\frac{dy}{dx} = \frac{1}{4} \times 2\left( x \pm a \right)\]

\[ \Rightarrow \frac{dy}{dx} = \frac{1}{2}\left( x \pm a \right)\]

Squaring both sides we get

\[ \Rightarrow \left( \frac{dy}{dx} \right)^2 = \left[ \frac{1}{2}\left( x \pm a \right) \right]^2 \]

\[ \Rightarrow \left( \frac{dy}{dx} \right)^2 = \frac{1}{4} \left( x \pm a \right)^2 \]

\[ \Rightarrow \left( \frac{dy}{dx} \right)^2 = y ............\left[\text{Using } \left( 1 \right) \right]\]

\[ \therefore y = \left( \frac{dy}{dx} \right)^2\]

Hence, the given function is the solution to the given differential equation.