Differential Equations: Common Mistakes and Fixes (1)

Question

Solve the differential equation $\dfrac{dy}{dx} + \dfrac{y}{x} = x^2$, given $y(1) = 2$.

Solution — Step by Step

Why This Works

The integrating factor $\mu(x)$ is constructed so that multiplying the ODE by $\mu$ makes the left side exactly $d(\mu y)/dx$. After that, both sides can be integrated directly.

The choice $\mu = e^{\int P\,dx}$ is forced by the product rule: we need $\mu' = \mu P$, which is exactly the differential equation $\mu$ itself satisfies.

Alternative Method

For this particular equation, we can also use the substitution $v = xy$ (since the structure suggests it). Then $dv/dx = y + x\,dy/dx = y + x(x^2 - y/x) = x^3$. Integrating gives $v = x^4/4 + C$, so $xy = x^4/4 + C$. Same answer faster.

Common Mistake