Question

Consider the differential equation :
\(\frac{dy}{dx}=\frac{y^3}{2(xy^2-x^2)}\)
Statement-1: The substitution \(z=y^2\) transforms the above equation into a first order homogenous differential equation.
Statement-2: The solution of this differential equation is \(y^2e^{-y^2}/x=C\).

• A. Both statements are false.
• B. Statement-1 is true and statement-2 is false.
• C. Statement-1 is false and statement-2 is true.
• D. Both statements are true.

Hint:

With \(z=y^2\): \(\frac{dz}{dx}=2y\frac{dy}{dx}=\frac{z^2}{x(z-x)}\), which is homogeneous. Then put \(z=vx\) and integrate.

Answer 1

The Correct Option is D

Let \(z=y^2\). Then \(\frac{dz}{dx}=2y\frac{dy}{dx}=\frac{y^4}{xy^2-x^2}=\frac{z^2}{xz-x^2}=\frac{z^2}{x(z-x)}\). The right side has degree 0 in \((x,z)\), so this is a first order homogeneous equation: Statement-1 is true.
Put \(z=vx\): \(v+x\frac{dv}{dx}=\frac{v^2}{v-1}\Rightarrow x\frac{dv}{dx}=\frac{v}{v-1}\Rightarrow\frac{v-1}{v}\,dv=\frac{dx}{x}\).
Integrate: \(v-\ln v=\ln x+c\Rightarrow\frac zx-\ln z=c\Rightarrow z=Ce^{z/x}\), i.e. \(y^2e^{-y^2/x}=C\).
This is the solution that Statement-2 intends (the printed \(y^2e^{-y^2}/x\) is the exponent \(-y^2/x\) with the fraction line lost in typesetting), so Statement-2 is true.
Answer: (4) Both statements are true.