Finding the general solution to a second-order ODE

[pat666]

Homework Statement

see attached

Homework Equations

The Attempt at a Solution

Question 7a) I can do but 7b I am not sure how to start.
I think x with a dot means differential with respect to time? need some help starting.

Thanks

 

[Dick]

Yes, x dot=dx/dt. It's separable.

 

[pat666]

Thanks Dick,
this question is confusing because x is y. anyway I have
x=sqrt(4t^2+8C)?THANKS

 

[Dick]

Thanks Dick,
this question is confusing because x is y. anyway I have
x=sqrt(4t^2+8C)?THANKS

That's not what I get from dx/dt=4*x*t. Can you explain how you got it?

 

[pat666]

from text:
dy/dx=f(x)g(y)
then int(1/g(y) .dy = int(f(x).dx int is integral
f(t)=t
g(x)=4x
then int(1/4x.dx)=int(t.dt)
x^2/8=t^2/2+C

so x=sqrt(4t^2+8C)

like I said this question is really confusing me because x is where y normally is.

 

[Dick]

int(1/(4x)*dx)=int(t*dt) is good. The right side is t^2/2+C. That's also good. But, the left side isn't x^2/8. Don't you get a log? int((1/x)*dx) is log(x) in my book.

 

[pat666]

Yes you do, my bad. so it should be ln(x)/4=t^2/2+C
so x=e^(2t^2+4c)??

 

[Dick]

Yes you do, my bad. so it should be ln(x)/4=t^2/2+C
so x=e^(2t^2+4c)??

That looks much better.

 

[pat666]

Sweet, THANKS