Differential eq y'(t) + y(t) * x'(t) = -2x(1)

Hi everyone. I'm having quite some problems with this differential equation:

y'(t) + y(t) * x'(t) = -2x(1)

where x(1) is x(t) where t=1

I'd like to integrate to get y(t). I've tried doing it using an integrating factor but I inevitably get stuck. Any help would be greatly appreciated. Thanks!

It appears that the integrating factor \(\displaystyle e^{x(t)}\) works.

\(\displaystyle \L\\y(t)+y(t)x'(t)=-2x(1)\)

\(\displaystyle \L\\\frac{dy}{dt}+y\frac{dx}{dt}=-2x(1)\)

Integrating factor would be:

\(\displaystyle \L\\e^{\int\frac{dx}{dt}}\) or \(\displaystyle e^{\int{x'(t)}\)

Integral of x'(t) is x(t)

So, you have \(\displaystyle e^{x(t)}\) as your integrating factor.

\(\displaystyle \L\\\frac{d}{dt}[ye^{x(t)}]=-2x(1)e^{x(t)}\)

Integrate giving:

\(\displaystyle \L\\ye^{x(t)}=\frac{-2x(1)e^{x(t)}}{x}+C\)

Solve for y:

\(\displaystyle \L\\y=\frac{-2x(1)}{x}+Ce^{-x(t)}\)

Hope I didn't make a boo-boo. Check it out.