Question

Form the differential equation for $y=(si{n}^{-1}x{)}^2+Aco{s}^{-1}x+B$ where A and B are arbitary constants, as its general solution.

Answer 1

Given that, $y=(si{n}^{-1}x{)}^2+Aco{s}^{-1}x+B$
On differentiating w.r.t. x, we get
$\frac{dy}{dx}=\frac{2si{n}^{-1}x}{\sqrt{1-x^2}}+\frac{(-A)}{\sqrt{1-x^2}}\Rightarrow \sqrt{1-x^2}\frac{dy}{dx}=2si{n}^{-1}x-A$
Again, differentiating w.r.t.x we get
$\sqrt{1-x^2}\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}.\frac{-2x}{2\sqrt{1+x^2}}=\frac{2}{\sqrt{1-x^2}}\Rightarrow (1-x^2)\frac{d^{2}y}{d{x}^2}-\frac{x}{\sqrt{1-x^2}}.\sqrt{1-x^2}\frac{dy}{dx}=2\Rightarrow (1-x^2)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}=2\Rightarrow (1-x^2)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}-2=0$
which is the required differential equation.