What is the general solution of the differential equation? # dy/dx=y(1+e^x) #

Question

Charles Arocha · Accepted Answer

# y = Ae^(x+e^x) #

We have: # dy/dx=y(1+e^x) # This is a first Order Separable Differential Equation, we can collect terms by rearranging the equation as follows # 1/y dy/dx=(1+e^x) # And now we can "separate the variables" to get # int \ 1/y \ dy = int \ 1+e^x \ dx # And integrating gives us: # ln|y| = x+e^x + C # # :. |y| = e^(x+e^x + C) # # :. |y| = e^(x+e^x) e^C # And as #e^x > 0 AA x in RR#, we can write the solution as: # :. y = Ae^(x+e^x) #

Luke Alvarez · Answer

undefined The general solution of the differential equation $ \frac{dy}{dx} = y(1 + e^x) $ can be found by separating variables and integrating.

First, we divide both sides by $ y(1 + e^x) $ to separate variables:

$$ \frac{dy}{y(1 + e^x)} = dx $$

Next, we integrate both sides:

$$ \int \frac{1}{y(1 + e^x)} dy = \int dx $$

$$ \int \frac{1}{y} \cdot \frac{1}{1 + e^x} dy = \int dx $$

$$ \int \frac{1}{y} dy = \int (1 + e^x) dx $$

$$ \ln|y| = x + e^x + C $$

Where $ C $ is the constant of integration.

Finally, we exponentiate both sides to solve for $ y $:

$$ |y| = e^{x + e^x + C} $$

$$ y = \pm e^{x + e^x + C} $$

Thus, the general solution of the differential equation is:

$$ y = Ce^{x + e^x} $$

Where $ C $ is an arbitrary constant.