dy/dx = e^(x+y) Solve | Solve dy/dx = e^(x-y)

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.

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