Good evening !

Thanks Guest, 

I just want to play too.    

$\frac{dx}{dt} * x^ 2 = e^{2t}\\ x^2\frac{dx}{dt} = e^{2t}\\ \int x^2\frac{dx}{dt}\;dt = \int e^{2t}\;dt\\ \int x^2\;dx = \int e^{2t}\;dt\\ \frac{x^3}{3} = \frac{ e^{2t}}{2}+c_1\\ x^3 = \frac{ 3e^{2t}}{2}+c_2\\ x=\left [\frac{ 3e^{2t}+c_3}{2}\right]^{1/3}\\$

Last line has been edited.

Solve the separable equation x(t)^2 ( dx(t))/( dt) = e^(2 t):
Integrate both sides with respect to t:
integral ( dx(t))/( dt) x(t)^2 dt  =   integral e^(2 t)  dt
Evaluate the integrals:
x(t)^3/3  =  e^(2 t)/2+c_1
Solve for x(t):
x(t) = -((-3/2)^(1/3) (e^(2 t)+2 c_1)^(1/3)) or x(t) = (3/2)^(1/3) (e^(2 t)+2 c_1)^(1/3) or x(t) = (-1)^(2/3) (3/2)^(1/3) (e^(2 t)+2 c_1)^(1/3)
Simplify the arbitrary constants:
Answer: | x(t) = -((-3/2)^(1/3) (e^(2 t)+c_1)^(1/3))           or x(t) = (3/2)^(1/3) (e^(2 t)+c_1)^(1/3) or x(t) = (-1)^(2/3) (3/2)^(1/3) (e^(2 t)+c_1)^(1/3)

Oups I feel super dumb haha thanks !

You might want to take another look at that last line Melody.

Bertie

Thanks Bertie,

I think it is better now ?