Q.The general solution of the differential equation $e^x\, dy + (y e^x + 2x)\, dx = 0$ is (A) $x e^y + x^2 = C$ (B) $x e^y + y^2 = C$ (C) $y e^x + x^2 = C$ (D) $y e^y + x^2 = C$

Concept: Integrating Factor Method – The equation is not directly separable, but rewriting it in standard linear form reveals an integrating factor.

Step 1: Rewrite in standard form.

Divide through by $dx$:

Step 2: Find the integrating factor.

Here $P(x)=1$, so integrating factor $\mu = e^{\int 1\,dx} = e^x$.

Step 3: Multiply and integrate.

Multiply both sides by $e^x$: …

We start with the equation:

A common first instinct is to try separating variables, but the presence of $y e^x$ mixed with $2x$ makes that impossible. Instead, notice that the equation is linear in $y$ — that is, it can be written in the form $\frac{dy}{dx} + P(x) y = Q(x)$. The Integrating Factor method is designed exactly for this: it turns the left side into the derivative of a product, making integration straightforward.

1. Rewrite in standard linear form

   Divide through by $dx$ and rearrange:

Then divide every term by $e^x$ (which is never zero):

Now it’s in the form $\frac{dy}{dx} + P(x) y = Q(x)$ with $P(x) = 1$ and $Q(x) = -2x e^{-x}$.

2. Find the Integrating Factor

   The integrating factor (I.F.) is $e^{\int P(x)\, dx}$. Here $\int 1\, dx = x$, so:

3. Multiply through by the I.F.

   Multiply the entire equation $\frac{dy}{dx} + y = -2x e^{-x}$ by $e^x$:

The left side is now exactly $\frac{d}{dx}(y e^x)$. Why? Because by the product rule:

So the equation becomes: …

Method: Rewrite $M\,dy+N\,dx=0$ as a linear equation in $y$

When a differential appears as $e^x\,dy+(\cdots)\,dx=0$, divide by $dx$ and check for linearity in $y$.

Steps

Step 1: Divide by $dx$.

Obtain $e^x\dfrac{dy}{dx}+(y e^x+2x)=0$.

Step 2: Reach standard linear form.

Divide by $e^x$: $\dfrac{dy}{dx}+y=-2x e^{-x}$, so $P=1$.

Step 3: Integrating factor $e^{x}$. …

Common Mistakes

Mistake 1: Trying to separate variables.

Why it's wrong: the $y e^x$ and $2x$ terms are mixed, so separation fails; the equation is linear in $y$. Correct approach: divide by $e^x$ to reach $\dfrac{dy}{dx}+y=-2x e^{-x}$.

Mistake 2: Sign error when moving $2x$ across. …