find the general solution of the differential equation with $e^x$

Question

$$\frac {dy}{dx} = (e^x - e^{-x})^2 $$

I am not sure how to find the general solution of this given problem. It is the e^x that is throwing me off .. any help would be greatly appreciated

Answer 1

HINT

Recall that

$$2\sinh x = e^x - e^{-x}$$

and$$ 2\sinh^2x=\cosh(2x)-1$$

Answer 2

Integrate directly $$\frac {dy}{dx} = (e^x - e^{-x})^2$$ $$y(x) = \int (e^x - e^{-x})^2dx$$ $$y(x) = \int (e^{2x} - 2+e^{-2x})dx$$

Answer 3

Expanding gives $$e^{2x}-2+e^{-2x}$$ this is easy to integrate.

Answer 4

RHS equals $ 4 \sinh^2 x $

Solution is $ y(x) = \int 4 \sinh^2 x dx = -2x + \sinh (2x) + c $

Answer 5

This is just a separable differential equation:

\begin{align} &\frac{dy}{dx}=(e^{x}-e^{-x})^{2}\\ &\Rightarrow dy=(e^{x}-e^{-x})(e^{x}-e^{-x})~dx\\ &\Rightarrow \int dy=\int (e^{2x}+e^{-2x}-2)~dx\\ &\Rightarrow y=\frac{1}{2}e^{2x}-\frac{1}{2}e^{-2x}-2x+C. \end{align}

As mentioned in other answers, you could also use the hyperbolic sine function.