Differential Equations: Edge Cases and Subtle Traps (7)

Question

Solve the differential equation $\dfrac{dy}{dx} + y = e^x$.

Solution — Step by Step

Why This Works

The integrating factor trick converts a linear first-order ODE into something we can integrate directly. After multiplying by $e^{\int P\, dx}$, the left side becomes the exact derivative of $y$ times the integrating factor. Then a single integration gives the solution.

This method works for any linear first-order ODE of the form above. The art lies in spotting that the equation is linear and identifying $P$ and $Q$ correctly.

Alternative Method

Find the homogeneous solution by solving $dy/dx + y = 0$: $y_h = Ce^{-x}$. Then guess a particular solution of the form $y_p = Ae^x$ (matching the right side). Substituting: $Ae^x + Ae^x = e^x$, so $A = 1/2$. General solution: $y = Ae^x/2 + Ce^{-x}$. Same answer.

This method (homogeneous + particular) is what we use for higher-order linear ODEs in JEE Advanced.

Common Mistake

Forgetting to multiply $Q(x)$ by the integrating factor before integrating. The correct integrand is IF $\times Q(x) = e^x \times e^x = e^{2x}$, not just $e^x$. Always multiply both sides by the IF before stepping into the integral.