Question

How do you reduce the order of a differential equation that has independent and dependent variables and no trivial solution?
In my case: x''+4tx'+x^2=tan(t)t^2

Answer 1

it is even analytically solvable?

Answer 2

That's exactly the problem.
I've been told that we need to replace x with two variables (y1 and y2). We then have 3 variables so I'm not really sure how it helps. We do however get a first order system.
(y1)=x
(y2)=x'
(y3)=x''=(y2)'
=>(y2)'+4t(y2)+(y1)^2=t^2tan(t)

Answer 3

That would technically be first order but we haven't really done anything with the equation.

Answer 4

If you can simplify it down to three first-order linear ODEs it can be solved using matrices and linear algebra.

Answer 5

It becomes a eigenvalue problem.