Runge Kutta for solving 2nd order ODE

[hsong9]

Homework Statement

(tw')' + tw = 0 with w(0) = 1, w'(0) = 0 on the interval [0,1]
by introducing the new variable v=tw' and considering the resulting first order differential system involving w and v
Report your computed solution (w_h(1),v_h(1)) for h=1/10.

Homework Equations

The Attempt at a Solution

x'(t) = f(x,t),
x(t₀) = x₀
I have no idea how I apply v'=tw into above formulas.
Could anyone show me one step?

 

[anathema]

Hello there,

I can help you with this problem. To introduce the new variable v=tw', we can rewrite the original equation as:

v' + w = 0

Now, we can consider the resulting first order differential system involving w and v:

w' = v/t
v' = -w

We can solve this system using the initial conditions w(0) = 1 and v(0) = 0. The solution for w(t) is:

w(t) = cos(t)

And the solution for v(t) is:

v(t) = -sin(t)

Now, to find the values of w and v at t=1/10, we can substitute t=1/10 into the above equations:

w(1/10) = cos(1/10) = 0.995
v(1/10) = -sin(1/10) = -0.0998

Therefore, the solution for (wh(1), vh(1)) is (0.995, -0.0998).

I hope this helps! Let me know if you have any further questions.