Derivation of equation of motion from various forces.

[Holychikenz]

Homework Statement

A particle of mass m is acted on by the forces as given below. Solve these equations
to find the motion of the particle in each case.

(a) F(x, t) = k(x + t2), with x = x0 and v = v0 = 0 when t = 0;
(b) F(x', t) = kx^2 x', with x = x0 and v = v0 = 0 when t = 0;
(c) F(x', t) = k(ax'+ t), with v = v0 when t = 0;
(d) F(x, x') = ax^2/x';
(e) F(x, x', t) = k(x + x't).

(a) I didn't have any trouble with as it was just a simple non homogeneous 2nd order ODE. The others to me seemed nonlinear and very difficult for some reason. I'm wondering how the variables which F is dependent on might affect the problems and if there is some method for solving these kinds of ODE's. Also my professor said that for each one to solve for x(t).

 

[Holychikenz]

In part (b) I get as far as solving for v(t), but to solve for x(t) I'm left with the integral of (1/(x^3 + x0^3)). Is very messy and ends up being tan-1(some function of x) + ln(another function of x). Because of that I can't solve for just x in terms of only t.

 

[gabbagabbahey]

For part (b), what is F(t=0)?...If the force on the particle is proportional to the speed, and its initial speed is zero, will the particle ever actually go anywhere?