Second Order differential equation involving chain rule

[Woolyabyss]

Homework Statement

Solve d^2x/dt^2 = (3x^3)/2

when dx/dt = -8 and x = 4 when t = 0



2. The attempt at a solution

v = dx/dt dv/dx = d^2/dx^2

d^2x/dt^2 = v(dv/dx) = (3x^3)/2

v dv = (3x^3)/2 dx

integrating and using limits and you get :

v^2/2 -32 = (3x^4)/8 - 96 ... 4v^2 = 3x^4 - 512

this is where I'm stuck I can take the square root of the 4v^2 but not the right hand side because of the three and the 512. I'm not sure if there is another technique I can use but finding the square root of both sides is the only way I was taught to do these problems and its the only example in the book.
Any help would be appreciated.

 

[Dick]

Homework Statement

Solve d^2x/dt^2 = (3x^3)/2

when dx/dt = -8 and x = 4 when t = 0



2. The attempt at a solution

v = dx/dt dv/dx = d^2/dx^2

d^2x/dt^2 = v(dv/dx) = (3x^3)/2

v dv = (3x^3)/2 dx

integrating and using limits and you get :

v^2/2 -32 = (3x^4)/8 - 96 ... 4v^2 = 3x^4 - 512

this is where I'm stuck I can take the square root of the 4v^2 but not the right hand side because of the three and the 512. I'm not sure if there is another technique I can use but finding the square root of both sides is the only way I was taught to do these problems and its the only example in the book.
Any help would be appreciated.

I think you are doing fine up till there. The square root is just sqrt(3x^4-512). But then if you want to find x(t) instead of v(x) you have to integrate something like dx/sqrt(3x^4-512), and that is going into elliptic integral country. I don't think you want to go there. I think you've gotten about as far as you can reasonably expect to get.

 

[Woolyabyss]

Alright Thanks I was able to get the question after it anyway I think I'll just leave this one for now.