Question

Solve the given differential equation by separation of variables. dy dx = e4x 5y

Answer 1

The solution to the given differential equation $\frac{dy}{dx} = e^{4x+5y}$ is , $\frac{e^{-5y}}{-5}=\frac{e^{4x}}{4} +c$, where c is constant of integration.

For given question,

We have been given a differential equation $\frac{dy}{dx} = e^{4x+5y}$

We know that for any real number a, m, n,

$a^{m + n} = a^m \times a^n$

⇒ dy/dx = $e^{4x}$ × $e^{5y}$

Separating the variables (x and its differential in one side and y and its differential in another side )

⇒ $\frac{1}{e^{5y}}$ dy = $e^{4x}$ dx

⇒ $e^{-5y}$ dy = $e^{4x}$ dx

Integrating on both the sides,

⇒ $\int e^{-5y}$ dy = $\int e^{4x}$ dx

We know that, $\int e^{ax}\, dx=\frac{e^{ax}}{a} +C$

⇒ $\int e^{4x}\, dx=\frac{e^{4x}}{4} +C$

and $\int e^{-5y}\, dy=\frac{e^{-5y}}{-5} +C$

So the solution is, $\frac{e^{-5y}}{-5}=\frac{e^{4x}}{4} +c$, where c is constant of integration.

Therefore, the solution to the given differential equation $\frac{dy}{dx} = e^{4x+5y}$ is , $\frac{e^{-5y}}{-5}=\frac{e^{4x}}{4} +c$, where c is constant of integration.

Learn more about the differential equation here:

brainly.com/question/14620493

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