Solve the linear first‑order differential equation dy/dx+2y=4x and find the particular solution that satisfies y(0)=1.

Solve the linear first‑order differential equation $\frac{d y}{d x} + 2 y = 4 x$ and find the particular solution that satisfies $y (0) = 1$ .

medium 2 marks Class 12 · Mathematics · Differential Equations

Answer

y = 2 x - 1 + 2 e^{- 2 x}

The correct answer is $y = 2 x - 1 + 2 e^{- 2 x}$ — Use initial condition $y (0) = 1$ : $1 = - 1 + C$ gives $C = 2$

Write the equation in standard form $y^{'} + P (x) y = Q (x)$ with $P (x) = 2$ , $Q (x) = 4 x$
Compute integrating factor $m u (x) = e^{in tP (x) d x} = e^{2 x}$
Multiply both sides by $m u (x)$ : $e^{2 x} y^{'} + 2 e^{2 x} y = 4 x e^{2 x}$
Recognise left side as derivative: $\frac{d}{d x} (e^{2 x} y) = 4 x e^{2 x}$
Integrate both sides: $e^{2 x} y = in t 4 x e^{2 x} d x + C$
Integrate $in t 4 x e^{2 x} d x$ by parts: $2 x e^{2 x} - e^{2 x}$
So $e^{2 x} y = 2 x e^{2 x} - e^{2 x} + C$
Divide by $e^{2 x}$ : $y = 2 x - 1 + C e^{- 2 x}$
Use initial condition $y (0) = 1$ : $1 = - 1 + C$ gives $C = 2$

Common mistake: Applying integrating factor incorrectly

Concept tested: Integrating factor for linear ODE

Published 2 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked