Q.The general solution of the differential equation $\dfrac{dy}{dx} = e^{x+y}$ is (A) $e^x + e^{-y} = C$ (B) $e^x + e^y = C$ (C) $e^{-x} + e^y = C$ (D) $e^{-x} + e^{-y} = C$

Concept: Separation of Variables — rewrite the equation so each variable appears on its own side, then integrate.

Step 1: Write $e^{x+y} = e^x e^y$, so

Step 2: Separate variables:

Step 3: Integrate both sides: …

Why Separation of Variables works here

The given equation is $\frac{dy}{dx} = e^{x+y}$. The right-hand side is a product of a function of $x$ and a function of $y$ — because $e^{x+y} = e^x \cdot e^y$. That’s the signal: whenever you can write $\frac{dy}{dx} = f(x) \cdot g(y)$, you can rearrange to get all $y$ on one side and all $x$ on the other, then integrate both sides. This is the method of separation of variables.

The trick is to treat $\frac{dy}{dx}$ as a fraction (which works in this context) and multiply both sides by $dx$ and by whatever is needed to isolate $y$ and $x$.

1. Rewrite the equation using the exponent rule $e^{x+y} = e^x \cdot e^y$, so

2. Separate the variables Multiply both sides by $dx$ and divide by $e^y$:

This is the separation step — every $y$ is on the left, every $x$ on the right.

3. Integrate both sides

The left integral is $\int e^{-y} dy = -e^{-y} + C_1$ (because the derivative of $-e^{-y}$ is $e^{-y}$).

The right integral is $\int e^x dx = e^x + C_2$.

So we have:

where $C = C_2 - C_1$ is a single arbitrary constant.

4. Rearrange to match the given options Multiply both sides by $-1$: $e^{-y} = -e^x - C$ …

Method: Separating an equation with $e^{x+y}$

Use this when an exponential mixes both variables in a sum, like $\dfrac{dy}{dx}=e^{x+y}$ — the key is that $e^{x+y}=e^x\,e^y$ is secretly a product, so it separates.

Steps

Step 1: Split the exponential into a product

revealing the separable form $\dfrac{dy}{dx}=e^x e^y$.

Step 2: Separate and integrate

Move $e^y$ across: $e^{-y}\,dy=e^x\,dx$, then integrate to $-e^{-y}=e^x+C$. …

Common Mistakes

Mistake 1: Treating $e^{x+y}$ as un-separable

Why it's wrong: $e^{x+y}=e^xe^y$ is a product, so the equation is separable; missing this leads to picking a wrong option. Correct approach: split the exponential, then separate.

Mistake 2: Sign error integrating $e^{-y}$ …