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**1. Homework Statement**

A particle of mass m is subject to a force F (x) = -kx. The initial position is

zero, and the initial speed is v

_{0}. Find x(t).

**2. Homework Equations**

F = m*v*dv/dx = -kx

v = dx/dt

**3. The Attempt at a Solution**

I'm new to differential equations, so please excuse me if I make any amateurish mistakes.

-kx = mv*dv/dx

After integrating:

-kx

^{2}/2 + c = mv

^{2}/2

v

^{2}= -kx

^{2}+ c

This is the part I'm unsure about, the inital values:

v

^{2}(0) = 0 + c = v

_{0}

^{2}

So c = v

_{0}

^{2}

v = (v

_{0}

^{2}- kx

^{2}/m)

^{1/2}

dx/v(x) = dt

After integrating: (a bit messy)

t = (m/k)

^{1/2}*arcsin(x*(k/v

_{0}

^{2}m)

^{1/2}) + c

c=0

Just wanted to know if I went wrong somewhere as I'm not convinced by the final answer.