Question

The general solution of the differential equation \(\frac{dy}{dx} + xy = 4x - 2y + 8\) is

• A. \(y = 4 - ce^{-\frac{(x+2)^2}{2}}\)
• B. \(y = 8 + ce^{-\frac{x^2}{2} - 2x}\)
• C. \(y = ce^{-(x+2)^2} + x\)
• D. \(y + 2x = ce^{-\frac{x}{2} - 2x}\)

Hint:

Always try to reduce the given differential equation to linear form and use the integrating factor method.

Answer 1

The Correct Option is A

Rewriting the given equation: \(\frac{dy}{dx} + xy = 4x - 2y + 8\). Rearranging terms: \(\frac{dy}{dx} + xy + 2y = 4x + 8\)
\(\Rightarrow \frac{dy}{dx} + y(x+2) = 4(x+2)\). Now use integrating factor (IF): \(IF = e^{\int (x+2)dx} = e^{\frac{x^2}{2} + 2x}\)
Multiplying through: \(\frac{d}{dx}(ye^{\frac{x^2}{2} + 2x}) = 4(x+2)e^{\frac{x^2}{2} + 2x}\)
Integrating both sides: \(y e^{\frac{x^2}{2} + 2x} = \int 4(x+2)e^{\frac{x^2}{2} + 2x} dx\)
Use substitution \(u = \frac{x^2}{2} + 2x\), so \(du = (x+2)dx\)
Solving the integral yields: \(y = 4 - ce^{-\frac{(x+2)^2}{2}}\)