Solving a 1st order nonlinear ODE

Question

Help with solving this nonlinear ODE analytically:

$$\frac{dx}{dt}=4x^2-16$$

I tried doing some kinds of variable substitutions but I was going nowhere.

The solution given is: $$\frac{2(x_0e^{16t}+x_0-2e^{16t}+2)}{-x_0e^{16t}+x_0+2e^{16t}+2}$$

Answer 1

Given equation is

$x'\:-4x^2+16=0$

$\Rightarrow \frac{1}{x^2-4}x'\:=4$

So it is the first-order ODE of the form

$$N(x)x'=M(t)$$ where $N(x)=\frac{1}{x^2-4}$ and $M(t)=4$

I hope you know how to solve Separable Equations (simple integration).

For hint check this

Answer 2

Hint.

Try to solve

$$ \frac{dx}{(2x-4)(2x+4)} = dt $$