Question

The general solution of the differential equation \((x+2y)^3\frac{dy}{dx} = y = 0, y>0\) is

• A. \(y = x^3 + cy\)
• B. \(x = y^3 + cy\)
• C. \(y(1 - xy) = cx\)
• D. \(x(1 - xy) = cy\)

Hint:

Try using substitution such as \(x = vy\) or \(y = vx\) when the equation involves nonlinear expressions in \(x\) and \(y\).

Answer 1

The Correct Option is B

We begin with \((x + 2y)^3 \frac{dy}{dx} = y\). Rearranging: \(\frac{dy}{dx} = \frac{y}{(x + 2y)^3}\)
Use substitution \(x = yv\), so \(dx = vdy + ydv\). Then transform and simplify. The equation becomes separable.
Integration eventually gives: \(x = y^3 + cy\)