Solve the given differential equation by separation of variables. dy/dx = e^(5x+4y).

Solution:

Given: Differential equation is dy/dx = e^(5x+4y).

dy/dx = e^(5x+4y).  {a^{m + n} = a^m × aⁿ}

dy/dx = e^5x × e^4y

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

⇒e^-4ydy = e^5xdx

Integrating on both the sides

⇒ \(\int e^{-4y}dy = \int e^{5x} dx\)

But \(\int e^{ax}dx\) = (e^ax/a) + c

⇒ \(\int e^{-4y}dx\) = (e^-4y/-4) + c and \(\int e^{5x}dx\) = (e^5x/5) + c

Therefore, (e^-4y/-4) = (e^5x/5) + c

Where c is constant of integration.

Solve the given differential equation by separation of variables. dy/dx = e^(5x+4y).

Summary:

The general solution of the differential equation by separation of variables. dy/dx = e^(5x+4y) is  (-e^-4y/4) = (e^5x/5) + c.