The general solution of dy/dx=e^{x+y} is equal to e^{x}+ e^{-y}= C, and the solution of dy/dx=e^{x-y} is given by e^{x} – e^{y}= C. Here C denotes an arbitrary constant. Let us learn to solve dy/dx = e^(x+y) and dy/dx = e^(x-y), and find their general solutions.

## General Solution of dy/dx=e^{x+y}

**Answer:** The general solution of the differential equation dy/dx=e^{x+y} is given by e^{x}+ e^{-y}= C, C is an integration constant.

*Proof:*

The given differential equation is

$\dfrac{dy}{dx}=e^{x+y}$

By the rule of indices, the above equation can be rewritten as follows:

$\dfrac{dy}{dx}$ = e^{x} e^{y}

Now, separating the variables we obtain that

e^{-y }dy = e^{x} dx

Integrating, ∫e^{-y }dy = ∫e^{x} dx + K

⇒ – e^{-y} =e^{x} + K

⇒ e^{x}+ e^{-y}= C, where C=-K.

Therefore, the solution of dy/dx=e^x+y is equal to e^{x}+ e^{-y}= C, where C denotes a constant.

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## General Solution of dy/dx=e^{x-y}

**Answer:** The solution of dy/dx=e^{x-y} is equal to e^{x}– e^{y}= C, C is any constant.

*Proof:*

We have:

$\dfrac{dy}{dx}=e^{x-y}$

⇒ $\dfrac{dy}{dx}$ = e^{x} e^{-y}

⇒ e^{y }dy = e^{x} dx

⇒ e^{x} dx – e^{y }dy = 0

Integrating both sides of the above equation, we obtain that

∫e^{x} dx – ∫e^{y }dy = C

⇒ e^{x}– e^{y}= C

So the solution of dy/dx=e^x-y is e^{x}– e^{y}= C, where C stands for an arbitrary constant.

## FAQs

### Q1: What is the general solution of dy/dx=e^{x+y}?

Answer: The general solution of dy/dx=e^{x+y} is e^{x}+e^{-y}=C.

### Q2: What is the general solution of dy/dx=e^{x-y}?

Answer: The general solution of dy/dx=e^{x-y} is e^{x}-e^{y}=C.