Problem 60
Find the general integral of the linear equation y'+y+x=0
Find the general integral of the linear equation $y'+y+x=0$.
Explanation of Method
Write in standard linear form $y'+y=-x$ and solve with the integrating factor $e^x$.
Step 1: Apply the integrating factor
With $\mu(x)=e^{\int1\,dx}=e^x$, multiplying through gives $\dfrac{d}{dx}(e^xy)=-xe^x$.
Step 2: Integrate
By parts, $\displaystyle\int -xe^x\,dx=e^x(1-x)+C$, so $e^xy=e^x(1-x)+C$, i.e.
\[\boxed{\,y=1-x+Ce^{-x}.}\]
Result
Final answer:
\[\boxed{\,y=1-x+Ce^{-x},\qquad C\in\mathbb{R}}\]
(homogeneous part $Ce^{-x}$ plus particular solution $1-x$).