# I am having trouble with a problem

• 1weasel
In summary, the second order nonlinear homogeneous DE has a first order equation in y', which can be solved because it is also separable. However, the integral is complicated.
1weasel
I am given the second order nonlinear homogeneous DE and I am supposed to find the equilibrium solutions for it.

u"+(u-1)u=0

I tried substuting u'=v but got stuck when I wasn't given initial values and couldn't solve for the constants after integrating.

1weasel said:
I am given the second order nonlinear homogeneous DE and I am supposed to find the equilibrium solutions for it.

u"+(u-1)u=0

I tried substuting u'=v but got stuck when I wasn't given initial values and couldn't solve for the constants after integrating.

If you use the substitution:

$$u=y$$
$$v=y'$$
$$\frac{dv}{du}=\frac{y''}{y'}$$

from which we have:

$$y=u$$
$$y'=v$$
$$y''=v\frac{dv}{du}$$

You end up with a first order DE:

$$vdv=(1-u)udu$$

Which is separable. The inverse substitution in the solution gives then a new first order equation in $y'$, which can be solved because it is also separable. However the integral is complicated. The end solution will be:

$$x=\int\frac{dy}{\sqrt{y^2-\frac{2}{3}y^3+K_1}}$$

Perhaps you have boundary conditions which can make life easier, although I think it will stay an unpleasant integral. Hope this helps a bit.

Last edited:
There seems to be something wrong with the latex generation, so here my edit:

I rewrote the equation as:
$$\frac{d^2y}{dx^2}+y(y-1)=0$$

I have tried to solve several times a Laplace transform with I.V.P's. There is what it looks like:

y" - y' -2y = F(t); where y(0) = 0, y'(0) = 0 and F(t) = 1 if 0<= t < 2
= t^2 + 1 if 2 <= t < 5
= t^2 + t if t => 5
I'm not having problem setting up but having trouble once I have everything on the right hand side. Unless I'm not using me brain the partial fractions look insane.

Solve the problem with F(t)= 1. Calculate the values of y and y' from that function at t= 2 and solve the initial value problem with F(t)= t2+ 1. Calculate the values of y and y' from that problem at t= 5 and solve the initial value problem with F(t)= t2+ t.

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