Stumped by this Differential Equation

[collegestudent0]

The problem is dy/dx=x+y, with the initial condition y(1)=1.

I put it in the linear form y'-y=x. Integrating factor is e^-x. I then get the general solution to be y=-x-1+C. Plugging in the initial condition, I find C to be 3, so the particular solution is y=2-x. However, when I try to check my answer:
-1=x+(2-x)
-1=2, which obviously isnt the case.

Help! What am I doing wrong here? Thank you!

 

[galactus]

The integrating factor IS \(\displaystyle e^{-x}\).

Multiply by the integrating factor we get:

\(\displaystyle \frac{d}{dx}e^{-x}y=xe^{-x}\)

Integrate both sides and get:

\(\displaystyle e^{-x}y=-xe^{-x}-e^{-x}+C\)

Divide through by \(\displaystyle e^{-x}\):

\(\displaystyle y=e^{x}-x-1\)

Use the initial condition to find C:

\(\displaystyle Ce^{1}-1-1=1\), solve for \(\displaystyle C=\frac{3}{e}\)

\(\displaystyle y=\frac{3}{e}e^{x}-x-1\)