Differential Equations: Exam-Pattern Drill (6)

Question

Solve the linear differential equation

$\frac{dy}{dx} + \frac{y}{x} = x^2, \quad x > 0$

with initial condition $y(1) = 2$.

Solution — Step by Step

Final answer: $y = x^3/4 + 7/(4x)$.

Why This Works

The integrating factor turns a linear ODE into an exact derivative. Once we recognise $\mu y$ as the product whose derivative we want, integration is mechanical.

The structure $\mu = e^{\int P\,dx}$ is engineered so that $(\mu y)' = \mu y' + \mu' y = \mu(y' + Py) = \mu Q$. That’s why the trick always works for linear ODEs.

Alternative Method

Recognise the homogeneous equation $y' + y/x = 0$ has solution $y_h = C/x$. Look for a particular solution $y_p = Ax^3$. Substituting: $3Ax^2 + Ax^2 = x^2$, so $A = 1/4$, giving $y_p = x^3/4$. Total: $y = x^3/4 + C/x$. Same answer via the homogeneous + particular method.

Common Mistake

Forgetting the constant of integration when integrating the integrating factor. Always include $C$ — it carries the initial condition. Skipping $C$ loses you 2 marks even if the rest is right.