Worked example y' − y = eˣ

Example — $y' - y = e^x$

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

Check: $y' = e^x(x+C) + e^x = e^x(x+C+1)$, therefore $y'-y = e^x(x+C+1)-e^x(x+C) = e^x$. ✓

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