Differential Equations Question - None

Question

The solution of the differential equation (x - y) dy - (x + y) dx = 0 is

Accepted Answer

Step 1: Rewrite the equation as \frac{dy}{dx} = \frac{x + y}{x - y}

Step 2: This is a homogeneous equation. Let y = vx, then \frac{dy}{dx} = v + x \frac{dv}{dx}

Step 3: Substituting: v + x \frac{dv}{dx} = \frac{x + vx}{x - vx} = \frac{1 + v}{1 - v}

Step 4: Simplifying: x \frac{dv}{dx} = \frac{1 + v}{1 - v} - v = \frac{1 + v^2}{1 - v}

Step 5: Separating variables: \frac{1 - v}{1 + v^2} dv = \frac{dx}{x}

Step 6: Integrating both sides: an^{-1} v - \frac{1}{2} \log(1 + v^2) = \log x + C

Step 7: Substituting back v = \frac{y}{x}: an^{-1} \frac{y}{x} - \frac{1}{2} \log(x^2 + y^2) = C

∴ Answer is (b).

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