Differential Equations: Conceptual Doubts Cleared (12)

Question

Solve the linear differential equation:

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

Find the particular solution that passes through $(1, 1)$.

Solution — Step by Step

Why This Works

The integrating factor $\mu = e^{\int P\,dx}$ is engineered so that multiplying through converts the left side into the derivative of a product. That step is the entire trick of linear ODEs — once the left becomes $\frac{d}{dx}(\mu y)$, integration finishes the job.

The general solution has two parts: the particular solution $x^3/5$ that responds to the forcing $x^2$, and the homogeneous solution $C/x^2$ that solves $dy/dx + 2y/x = 0$. Initial conditions fix the constant $C$.

Alternative Method

Recognise that the homogeneous part $dy/dx = -2y/x$ is separable: $\ln|y| = -2\ln|x| + c$, giving $y_h = C/x^2$. Then guess $y_p = Ax^3$ as a particular solution and find $A = 1/5$ by substituting. The general solution is the sum.

Common Mistake