By using a suitable substitution, bring the equation…

By using a suitable substitution, bring the equation $y'=(x-y)^2+1$ to a form where the variables can be separated, and then integrate it.

Explanation of Method

Since the equation depends on $x$ and $y$ only through $x-y$, substitute $v=x-y$ to reduce it to a separable equation in $v$ and $x$.

With $v=x-y$, $y=x-v$ and $y'=1-v'$. Substituting into $y'=(x-y)^2+1$:

\[1-v'=v^2+1 \quad\Longrightarrow\quad v'=-v^2.\]

\[\frac{dv}{v^2}=-dx \quad\Longrightarrow\quad -\frac1v=-x+C \quad\Longrightarrow\quad v=\frac{1}{x-C}.\]

Since $v=x-y$, we get $x-y=\dfrac{1}{x-C}$, so (relabelling $c=-C$)

\[\boxed{\,y=x-\frac{1}{x+c}.}\]

Final answer:

\[\boxed{\,y=x-\dfrac{1}{x+c},\qquad x\neq -c.}\]