If $\mathrm{y}=\sin \left(3{\sin }^{-1}\mathrm{x}\right)$ and a, b, c, d are such that $\mathrm{a}\left(1-{\mathrm{x}}^2\right)\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}+\mathrm{bx}\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{cy}=\mathrm{d}$. Then a + b + c value is . . . .

${\sin }^{-1}\mathrm{y}=3{\sin }^{-1}x$

 Differentiate w.r.to x

 $$\begin{gathered} \frac{{\mathrm{y}}_1}{\sqrt{1-{\mathrm{y}}^2}}=\frac{3}{\sqrt{1-{\mathrm{x}}^2}} \\ \left(1-x^2\right){\mathrm{y}}_1^2=9\left(1-{\mathrm{y}}^2\right) \\ Again differentiate w.r.to.x on both sides \\ (1-x^2)\left(2y_1{y}_2\right)+{{y_1}^2}_{ }(-2x)=9(-2y)y_1 \\ \Rightarrow (1-x^2)y_2-x{y}_1+9y=0 \\ By comparision a=1,b=-1,c=9,d=0 \\ Now a+b+c=1-1+9=9 \end{gathered}$$