Q.Solve the differential equation $y e^{x/y}\, dx = \left(x e^{x/y} + y^2\right) dy\ (y \neq 0)$.

The exponent $x/y$ signals the substitution $v=\frac{x}{y}$, i.e. $x=vy$, so $dx=v\,dy+y\,dv$ and $e^{x/y}=e^{v}$.

Put these into $y e^{x/y}\,dx=(x e^{x/y}+y^2)\,dy$:

Expand: $vy e^{v}\,dy+y^2 e^{v}\,dv=vy e^{v}\,dy+y^2\,dy$. The $vy e^{v}\,dy$ terms cancel, leaving …

Reading the equation

The awkward part is $e^{x/y}$, whose exponent is the ratio $x/y$. That is the cue to make the ratio a new variable: let $v=\dfrac{x}{y}$, so $x=vy$.

Substitute $x=vy$

Then $dx=v\,dy+y\,dv$ and $e^{x/y}=e^{v}$. The equation becomes

Simplify

Expand the left side:

The term $vy e^{v}\,dy$ appears on both sides and cancels:

Since $y\neq 0$, divide by $y^2$:

The variables are now separated.

Integrate …

Method: Substitution suggested by an $e^{x/y}$ term

When a stubborn expression like $e^{x/y}$ appears, make its exponent the new variable — here $v=\dfrac{x}{y}$, i.e. $x=vy$.

Steps

Step 1: Set $v=\dfrac{x}{y}$ so $x=vy$.

Then $dx=v\,dy+y\,dv$ and $e^{x/y}=e^{v}$.

Step 2: Substitute everywhere.

Replace $x$, $dx$ and $e^{x/y}$ in the equation.

Step 3: Cancel and separate. …

Common Mistakes

Mistake 1: Not letting the $e^{x/y}$ exponent guide the substitution.

Why it's wrong: the ratio $x/y$ inside the exponential signals $v=\tfrac{x}{y}$, i.e. $x=vy$; a generic $y=vx$ leaves $e^{1/v}$ and does not simplify. Correct approach: substitute $x=vy$ so $e^{x/y}=e^{v}$.

Mistake 2: Differentiating $x=vy$ incorrectly.

Why it's wrong: $dx=v\,dy+y\,dv$ by the product rule; dropping $y\,dv$ loses the equation. Correct approach: expand $dx$ as a product differential. …