Differential Equations
Homogeneous Differential Equations
Premium Question
Grade 12
Question:
The general solution of $(x + y)\frac{dy}{dx} = y - x$ is:
(1) $\ln(x^2 + y^2) + \tan^{-1}\left(\frac{y}{x}\right) = C$
(2) $\frac{1}{2}\ln(x^2 + y^2) + \tan^{-1}\left(\frac{y}{x}\right) = C$
(3) $\frac{1}{2}\ln(x^2 + y^2) - \tan^{-1}\left(\frac{y}{x}\right) = C$
(4) $\ln(x^2 + y^2) - 2\tan^{-1}\left(\frac{y}{x}\right) = C$
Step-by-Step Solution
Key Concept: Substitute $y = vx$ to reduce this homogeneous equation to a separable form, then split the resulting integrand $\frac{v + 1}{1 + v^2}$ into $\frac{v}{1 + v^2} + \frac{1}{1 + v^2}$ before integrating.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)