Consider the differential equation:

(du/dt)=-u^2(t^3-t)

a) Find the general solution to the above differential equation. (Write the answer in a form such that its numerator is 1 and its integration constant is C).

u=?

b) Find the particular solution of the above differential equation that satisfies the condition u=4 at t=0.

u=?

-du/u^2 = (t^3 - t) dt --->

1/u = 1/4 t^4 - 1/2 t^2 + c ---->

u = 1/[1/4 t^4 - 1/2 t^2 + c]

u=4 at t=0 ---> c = 1/4

du/u^2 = -(t^3 -t ) dt

1/u = (1/4)t^4 - (1/2)t^2 + constant
1/u = -t^2 (.5 -.25 t^2) + constant
u = -1/[t^2(.5 - .25 t^2) + C]

4 = -1/C so C = -4

u = -1/[t^2(.5 - .25 t^2) - 4]

I forgot a - sign - use his :)

Thanks so much. I've been struggling with that thing for days!

Oh - actually we agree exactly

To find the general solution to the given differential equation, we need to separate variables and integrate.

a) Let's start by separating the variables in the equation:

(du/u^2) = -(t^3 - t) dt

Now, we can integrate both sides of the equation. The integral of 1/u^2 with respect to u is -1/u, and the integral of -(t^3 - t) with respect to t is -(1/4)t^4 + (1/2)t^2. Integrating both sides, we have:

-1/u = -(1/4)t^4 + (1/2)t^2 + C

To put the answer in the requested form (numerator as 1, and integration constant as C), we can multiply through by -1:

1/u = (1/4)t^4 - (1/2)t^2 - C

Now, we can take the reciprocal of both sides to get the general solution:

u = 1 / ((1/4)t^4 - (1/2)t^2 - C)

Therefore, the general solution to the given differential equation is u = 1 / ((1/4)t^4 - (1/2)t^2 - C).

b) To find the particular solution that satisfies the condition u = 4 at t = 0, we can substitute these values into the general solution and solve for the integration constant C:

4 = 1 / ((1/4)(0)^4 - (1/2)(0)^2 - C)

Simplifying the equation:

4 = 1 / (0 - 0 - C)
4 = 1 / (-C)

Multiplying both sides of the equation by (-C), we get:

-4C = 1

Dividing both sides by -4, we find:

C = -1/4

Substituting this value back into the general solution:

u = 1 / ((1/4)t^4 - (1/2)t^2 - (-1/4))

Simplifying further:

u = 1 / ((1/4)t^4 - (1/2)t^2 + 1/4)

Therefore, the particular solution of the given differential equation that satisfies the condition u = 4 at t = 0 is u = 1 / ((1/4)t^4 - (1/2)t^2 + 1/4).