Question

Solution of the differential equation \(\frac{dy}{dx} = e^{3x-2y} + x^2 e^{-2y}\) is ______

(a) \( \frac{e^{2y}}{3} = \frac{e^{3x}}{3} + \frac{x^2}{2} + c\)

(b) \( \frac{e^{3y} (e^{2x}+x^3)}{6} + c\)

(c) \(\frac{e^{2y} (e^{3x}+x^3)}{6} + c\)

(d) \( \frac{e^{2y}}{2} = \frac{e^{3x}}{3} + \frac{x^3}{3} + c\)

This question was addressed to me in an interview for internship.

This interesting question is from Separable and Homogeneous Equations in chapter Ordinary Differential Equations – First Order & First Degree of Engineering Mathematics

Answer 1

Correct option is (d) \( \frac{e^{2y}}{2} = \frac{e^{3x}}{3} + \frac{x^3}{3} + c\)

For explanation: \(\frac{dy}{dx} = e^{3x-2y} + x^2 e^{-2y}\)

\(\frac{dy}{dx} = e^{-2y} (e^{3x} + x^2)\)

separating the variable

 e^2y dy = (e^3x+x^2)dx…..integrating

∫ e^2y dy =∫ (e^3x+x^2)dx

\( \frac{e^{2y}}{2} = \frac{e^{3x}}{3} + \frac{x^3}{3} + c\).