Find the general integral of the homogeneous equation…
Find the general integral of the homogeneous equation $y'=e^{x/y}+\dfrac{x}{y}$.
Explanation of Method
The right side depends only on $x/y$, hence also only on $t=y/x$ (since $x/y=1/t$), so the equation is homogeneous. The standard substitution $y=tx$ reduces it to a separable equation whose integral is non-elementary and must be left implicit.
Step 1: Substitute $y=tx$
With $t=y/x$, so $x/y=1/t$, the equation reads $y'=e^{1/t}+1/t$. Since $y=tx$ gives $y'=t+xt'$,
\[t+x\frac{dt}{dx}=e^{1/t}+\frac1t \quad\Longrightarrow\quad x\frac{dt}{dx}=e^{1/t}+\frac1t-t.\]
Step 2: Separate variables
\[\frac{dt}{\,e^{1/t}+\dfrac1t-t\,}=\frac{dx}{x}.\]
Step 3: Integrate
The left-hand integral has no elementary antiderivative, so the general integral is left implicit:
\[\boxed{\,\int\frac{dt}{e^{1/t}+\frac1t-t}=\ln|x|+C,\qquad t=\frac{y}{x}.}\]
Result
Final answer:
\[\boxed{\,\int\frac{dt}{e^{1/t}+\tfrac1t-t}=\ln|x|+C,\qquad t=\dfrac{y}{x}}\]
(no elementary closed form exists).