Question

Show that Ax² + By² = 1 is a solution of the differential equation x $\left\{y\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2\right\}=y\frac{dy}{dx}$.

Answer 1

We have,
$A{x}^2+B{y}^2=1$ ...(1)
Differentiating both sides of (1) with respect to $x$, we get
$2Ax+2By\frac{dy}{dx}=0$ ...(2)
Differentiating both sides of (2) with respect to $x$, we get

$$\begin{gathered} 2A+2B{\left(\frac{dy}{dx}\right)}^2+2By\frac{d^{2}y}{d{x}^2}=0 \\ \Rightarrow 2B\left[y\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2\right]=-2A \\ \Rightarrow \left[y\frac{dy}{dx}+{\left(\frac{dy}{dx}\right)}^2\right]=-\frac{2A}{2B} \\ \Rightarrow \left[y\frac{dy}{dx}+{\left(\frac{dy}{dx}\right)}^2\right]=-\left(-\frac{y}{x}\frac{dy}{dx}\right)\left[\text{Using}\left(2\right)\right] \\ \Rightarrow x\left[y\frac{dy}{dx}+{\left(\frac{dy}{dx}\right)}^2\right]=y\frac{dy}{dx} \end{gathered}$$
Hence, the given function is the solution to the given differential equation.