Question

Form the differential equation having y = (sin–1x)2
+ Acos–1x + B, where A and B
are arbitrary constants, as its general solution

Answer 1

Answer on Question #84395 – Math – Differential Equations

Question

Form the differential equation having $y = (\sin^{-1}x)^2 + A\cos^{-1}x + B$, where $A$ and $B$ are arbitrary constants, as its general solution.

Solution

We have

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

Differentiating the given function w.r.t. $x$ successively, we get

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

Hence

$$\sqrt{1 - x^2} \frac{dy}{dx} = 2 \sin^{-1} x - A$$

On again differentiating w.r.t. $x$, we get

$$\sqrt{1 - x^2} \frac{d^2 y}{d x^2} + \left(-\frac{2x}{2\sqrt{1 - x^2}}\right) \frac{dy}{dx} = 2 \left(\frac{1}{\sqrt{1 - x^2}}\right)$$ $$(1 - x^2) \frac{d^2 y}{d x^2} - x \frac{dy}{dx} - 2 = 0$$

This is the required differential equation.

Answer: $(1 - x^2) \frac{d^2 y}{d x^2} - x \frac{dy}{dx} - 2 = 0$

Answer provided by https://www.AssignmentExpert.com