What is the general solution of the differential equation? $\frac{d y}{d x} = y (1 + e^{x})$ $dy/dx=y(1+e^x)$

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Answer 1 · 2017-06-12T15:02:03

$y = A e^{x + e^{x}}$ $y = Ae^(x+e^x)$

We have:

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

This is a first Order Separable Differential Equation, we can collect terms by rearranging the equation as follows

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

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:

$:. y = Ae^(x+e^x)$

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