Second-Order Nonlinear Differential Equation

sav26

New member
Joined
Nov 21, 2020
Messages
3
Hi there can someone please help me with this differential equation, I'm having trouble solving it
[MATH] \begin{cases} y''(t)=-\frac{y(t)}{||y(t)||^3} \ , \forall t >0 \\ y(0)= \Big(\begin{matrix} 1\\0\end{matrix} \Big) \ \text{and} \ y'(0)= \Big(\begin{matrix} 0\\1\end{matrix} \Big) \end{cases} \\ y(t) \in \mathbb{R}^2 \ \forall t [/MATH]
Thanks in advance ^^
 
Last edited:
I don't know about solving it in general, but if you insist that [MATH]\|y(t)\|^3=1[/MATH]then a solution is easily found as

[MATH] y(t) = \begin{pmatrix}\cos(t)\\ \sin(t)\end{pmatrix}[/MATH]
 
Thanks for your response, but i can't really declare that [MATH] ||y(t)||^3=1 [/MATH] (':
 
You can handwave it, if you express your vector y as x1 and x2 and show that you can formulate a new system of equations that does not contain the (x12+x22)3/2 (a bit of algebra).

This system can be "solved" by inspection or maybe with another method.
 
A note, since the equation is nonlinear, I'm not actually sure the solution is unique and I don't remember/have knowledge of the existence and uniqueness theorems of equations of this type.
 
Top