$\begin{align}  & y=\sin (m{{\sin }^{-1}}x) \\ & \text{Let }u=m{{\sin }^{-1}}x \\ & y=\sin u,\text{  }\frac{dy}{du}=\cos u \\ & \frac{du}{dx}=\frac{m}{\sqrt{1-{{x}^{2}}}} \\ & \frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}=\cos u\times \frac{m}{\sqrt{1-{{x}^{2}}}} \\ & \frac{dy}{dx}=\frac{m\cos u}{\sqrt{1-{{x}^{2}}}} \\ & \frac{dy}{dx}=\frac{m\cos (m{{\sin }^{-1}}x)}{\sqrt{1-{{x}^{2}}}} \\ & \sqrt{1-{{x}^{2}}}\frac{dy}{dx}=m\cos (m{{\sin }^{-1}}x) \\ & \sqrt{1-{{x}^{2}}}\frac{{{d}^{2}}y}{d{{x}^{2}}}+\frac{1}{2}\left( \frac{dy}{dx} \right)\frac{-2x}{\sqrt{1-{{x}^{2}}}}=-m\sin (m{{\sin }^{-1}}x)\times \frac{m}{\sqrt{1-{{x}^{2}}}} \\ & \sqrt{1-{{x}^{2}}}\frac{{{d}^{2}}y}{d{{x}^{2}}}-\frac{x}{\sqrt{1-{{x}^{2}}}}\frac{dy}{dx}=\frac{-{{m}^{2}}\sin (m{{\sin }^{-1}}x)}{\sqrt{1-{{x}^{2}}}} \\ & \text{Multiply through by }\sqrt{1-{{x}^{2}}} \\ & (1-{{x}^{2}})\frac{{{d}^{2}}y}{d{{x}^{2}}}-\frac{dy}{dx}=-{{m}^{2}}\sin (m{{\sin }^{-1}}x) \\ & (1-{{x}^{2}})\frac{{{d}^{2}}y}{d{{x}^{2}}}=\frac{dy}{dx}-{{m}^{2}}(m{{\sin }^{-1}}x) \\ & (1-{{x}^{2}})\frac{{{d}^{2}}y}{d{{x}^{2}}}=\frac{dy}{dx}-{{m}^{2}}y \\\end{align}$