Problem 61
Find the general integral of the linear equation y'-y=e^x
Find the general integral of the linear equation $y'-y=e^x$.
Explanation of Method
This is linear in standard form $y'+P(x)y=Q(x)$ with $P=-1$, $Q=e^x$. Multiply by the integrating factor $\mu(x)=e^{-x}$.
Step 1: Apply the integrating factor
With $\mu(x)=e^{-x}$, the equation becomes $\dfrac{d}{dx}(e^{-x}y)=e^{-x}\cdot e^x=1$.
Step 2: Integrate
Integrating, $e^{-x}y=x+C$, so
\[\boxed{\,y=xe^x+Ce^x.}\]
Result
Final answer:
\[\boxed{\,y=xe^x+Ce^x,\qquad C\in\mathbb{R}}\]