Question

Differential equation having
$y=({\sin }^{-1}x{)}^2+A({\cos }^{-1}x)+B$ where $A,B$ are arbitrary constants as general solution is

• A. $(1-x^2)y_2-x{y}_1=2$
• B. $(1-x^2)y_2-x{y}_1=1$
• C. $(1-x)y_2-x{y}_1=2$
• D. $(1-x^2)y_2-x{y}_1=4$

Answer 1

The correct option is A $(1-x^2)y_2-x{y}_1=2$

$y=({\sin }^{-1}x{)}^2+A({\cos }^{-1}x)+B$

$\Rightarrow \frac{dy}{dx}=2\left({\sin }^{-1}x\right)\frac{1}{\sqrt{1-x^2}}+A\frac{-1}{\sqrt{1-x^2}}$

$\Rightarrow \sqrt{1-x^2}{y}_1=2\left({\sin }^{-1}x\right)-A$

$\Rightarrow \sqrt{1-x^2}{y}_2+y_1.\frac{1}{2\sqrt{1-x^2}}(-2x)=\frac{2}{\sqrt{1-x^2}}$

$\Rightarrow (1-x^2)y_2-x{y}_1=2$