Reduction of second order differential equation $u''=2u^3$

Question

Given the differential equation $u''=2u^3$, what method of reduction can I use to make it easier to solve?

The reduction order method requires a solution to be known and I am unsure on where to go from here.

Answer 1

Hint:$$u''=2u^3\to\\2u'u''=4u'u^3\to\\(u')^2=u^4+C_1\to\\u'=\pm\sqrt{u^4+C_1}$$can you finish now?

Answer 2

Multiply by $2u'$ to get

$$2u'u''=4u^3u'$$ and by integration

$$u'^2=u^4+C.$$

Then a separable equation

$$\frac1{\sqrt{u^4+C}}=\pm dt.$$

Unfortunately, this one is difficult.

Answer 3

Hint

$$u''=2u^3$$ $$\frac {d u'}{dx}=2u^3$$ $$\frac {d u'}{du}\frac {du}{dx}=2u^3$$ $$\frac {d u'}{du} u'=2u^3$$ $$\int u'{d u'} = 2\int u^3 du$$ $$ \frac {(u')^2} 2 = \frac {u^4} 2+K$$ $$ {(u')^2} = {u^4} +K$$

The integration of this equation is not that simple.