By a suitable substitution or differentiation, reduce…

By a suitable substitution or differentiation, reduce $y'-1=e^{x+2y}$ to linear form and find its general integral.

Step 1: Isolate $y'$

$y'=1+e^{x}e^{2y}$.

Step 2: Substitute $u=e^{-2y}$

Let $u=e^{-2y}$, so $u'=-2uy'$, i.e. $y'=-u'/(2u)$; also $e^{2y}=1/u$. Substituting:

\[-\frac{u'}{2u}=1+\frac{e^x}{u}.\]

Multiply by $-2u$:

\[u'=-2u-2e^x\quad\Longrightarrow\quad u'+2u=-2e^x.\]

Step 3: Solve the linear equation in $u$

With integrating factor $\mu(x)=e^{2x}$: $\dfrac{d}{dx}(ue^{2x})=-2e^{3x}$, so $ue^{2x}=-\dfrac23e^{3x}+C$, i.e.

\[u=-\frac23 e^{x}+Ce^{-2x}.\]