Question

The solution of the differential equation $\frac{x \frac{dy}{dx} - y}{\sqrt{x^2 - y^2}} = 10x^2$ is:

• A. $\sin^{-1} \left( \frac{y}{x} \right) - 5x^2 = C$
• B. $\sin^{-1} \left( \frac{y}{x} \right) = 10x^2 + C$
• C. $\frac{y}{x} = 5x^2 + C$
• D. $\sin^{-1} \left( \frac{y}{x} \right) = 10x^2 + Cx$
• E. $\sin^{-1} \left( \frac{y}{x} \right) + 5x^2 = C$

Hint:

The expression $\frac{xdy-ydx}{x^2}$ always suggests the substitution $v=\frac{y}{x}$.

Answer 1

The Correct Option is A

Concept:
Notice the standard differential form: \[ d\left(\frac{y}{x}\right)=\frac{x\,dy-y\,dx}{x^2} \] This suggests the substitution: \[ v=\frac{y}{x} \]

Step 1: Rewrite the given equation.
Given: \[ \frac{x\,dy-y\,dx}{\sqrt{x^2-y^2}}=10x^2\,dx \] Factor $x^2$ inside the square root: \[ \sqrt{x^2-y^2} = x\sqrt{1-\left(\frac{y}{x}\right)^2} \] Substitute: \[ \frac{x\,dy-y\,dx}{x\sqrt{1-(y/x)^2}} = 10x^2\,dx \] Divide numerator and denominator by $x^2$: \[ \frac{\dfrac{x\,dy-y\,dx}{x^2}} {\sqrt{1-(y/x)^2}} = 10x\,dx \] Using: \[ d\left(\frac{y}{x}\right) = \frac{x\,dy-y\,dx}{x^2} \] we get: \[ \frac{d(y/x)} {\sqrt{1-(y/x)^2}} = 10x\,dx \]

Step 2: Integrate both sides.
\[ \int \frac{d(y/x)} {\sqrt{1-(y/x)^2}} = \int 10x\,dx \] Using: \[ \int \frac{dv}{\sqrt{1-v^2}} = \sin^{-1}(v) \] we obtain: \[ \sin^{-1}\left(\frac{y}{x}\right) = 5x^2+C \] Final Answer:
\[ \boxed{ \sin^{-1}\left(\frac{y}{x}\right)-5x^2=C } \]