Good evening !
What would be the answer of : dx/dt * x2 = e2t
It would be great of you to answer !
Thks and happy new year
+118609 Thanks Guest,
I just want to play too. 
\(\frac{dx}{dt} * x^ 2 = e^{2t}\\ x^2\frac{dx}{dt} = e^{2t}\\ \int x^2\frac{dx}{dt}\;dt = \int e^{2t}\;dt\\ \int x^2\;dx = \int e^{2t}\;dt\\ \frac{x^3}{3} = \frac{ e^{2t}}{2}+c_1\\ x^3 = \frac{ 3e^{2t}}{2}+c_2\\ x=\left [\frac{ 3e^{2t}+c_3}{2}\right]^{1/3}\\\)
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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)
+118609 Thanks Guest,
I just want to play too.
\(\frac{dx}{dt} * x^ 2 = e^{2t}\\ x^2\frac{dx}{dt} = e^{2t}\\ \int x^2\frac{dx}{dt}\;dt = \int e^{2t}\;dt\\ \int x^2\;dx = \int e^{2t}\;dt\\ \frac{x^3}{3} = \frac{ e^{2t}}{2}+c_1\\ x^3 = \frac{ 3e^{2t}}{2}+c_2\\ x=\left [\frac{ 3e^{2t}+c_3}{2}\right]^{1/3}\\\)
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