I have been having some trouble with the following question - i'm not sure how the substitution works and how to change back at the end.
Any help greatly appreciated!
Show that posing x = t +1/z changes the equation x' = x^2 - tx +1 to a linear equation. Then Solve.
The substitution is in fact x = t + 1/z. I have substituted in but came across a few difficulties. Firstly, i was unsure as to how to calculate (t + 1/z)'. I came up with t'. Is this correct?
Assuming it is, the ODE yielded t' - (1/z)t = 1 + 1/z^2.
I tried to solve this and ended up with t(z) = zln|z| - 1/2z.
Does this seem correct. My major problem was how to change it back to get a solution x(t).
THanks so much for your help.
Ok, then I got z' +tz = -1.
I solved z' + tz = 0 first and got Ce^(-t^2/2)
I couldn't seem to find a particular solution so I varied the constant and ended up with the general solution z(t) = g(t)/g'(t) + c, where g(t) is the integral of e^(t^2/2)
Does this seem correct? The question asks me to express this in terms of the initial conditions (t(0), x(0) ). How do I translate it back to a solution for x(t)?
Thanks again for all your help.