You can't.how can I transform this into y wrt x and z wrt x without going to second order
y = Y(x) and z = Z(x) simultaneously satisfy the system of equation:
y' = z, z' = x^3 (y = z)
The problem gives you initial conditions and initial guesses (for successive approximation). But my question is how can I transform this into y wrt x and z wrt x without going to second order. Because the formula that I know for doing approximation involves only two variable and only to the first order.
Why not just integrate the z equation first, obtaining
and then integrate the y equation, getting
Kinda simplistic, but it gets the job done, doesn't it? That is, if you're trying to solve the DE exactly, here's the exact solution. If you're doing successive approximations, that's something else.
You might try matrix methods. The system becomes
The trivial solution obviously works here. Do you have any initial conditions? You might have a hard time coming up with a fundamental matrix.
Any particular reason you have to avoid the second-order method?