Question

If $x=\sin \left(\frac{1}{a}\log y\right)$, show that (1 − x²)y₂ − xy₁ − a²y = 0.

Answer 1

Here,
$$\begin{gathered} x=\sin \left(\frac{1}{a}\log y\right) \\ \Rightarrow \frac{1}{a}\log y={\sin }^{-1}x \\ \Rightarrow y=e^{a{\sin }^{-1}x} \\ \mathrm{Differentiating}\mathrm{w}.\mathrm{r}.\mathrm{t}.x,\mathit{}\mathrm{we}\mathrm{get} \\ {y}_1=e^{a{\sin }^{-1}x}\times \frac{a}{\sqrt{1-x^2}} \\ \Rightarrow y_1=\frac{ay}{\sqrt{1-x^2}} \\ \mathrm{Differentiating}\mathrm{again}\mathrm{w}.\mathrm{r}.\mathrm{t}.x,\mathrm{we}\mathrm{get} \\ {y}_2=\frac{a{y}_1\sqrt{1-x^2}+{\displaystyle \frac{xay}{\sqrt{1-x^2}}}}{\left(1-x^2\right)} \\ \Rightarrow y_2=\frac{a{y}_1\left(1-x^2\right)+{\displaystyle xay}}{\left(1-x^2\right)\sqrt{1-x^2}} \\ \Rightarrow y_2=\frac{a{y}_1}{\sqrt{1-x^2}}+\frac{xay}{\left(1-x^2\right)\sqrt{1-x^2}} \\ \Rightarrow y_2=\frac{a^{2}y}{1-x^2}+\frac{x{y}_1}{\left(1-x^2\right)} \\ \Rightarrow \left(1-x^2\right)y_2-x{y}_1-a^{2}y=0 \end{gathered}$$