Worked exercise — y' − y = eˣ

Problem

Solve $y'-y=e^x$ (first-order linear non-homogeneous).

Solution

Here $p(x)=-1$, $P(x)=\int p\,dx=-x$, $q(x)=e^x$. Using the solving formula: $y = e^{x}\left[\int e^x\cdot e^{-x}\,dx + C\right] = e^x\left[\int 1\,dx + C\right] = e^x(x + C).$ $\boxed{y(x) = e^x(x + C)}$

Links

Topics: Differential equations
Concepts: First-order linear ODE
Skills: Integrating · Solving equations · Using formulae
Exercise type: Equation solving