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
\(\displaystyle
\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
\)

Thanks in advance ^^
 
Last edited:

Romsek

Senior Member
Joined
Nov 16, 2013
Messages
1,006
I don't know about solving it in general, but if you insist that \(\displaystyle \|y(t)\|^3=1\)
then a solution is easily found as

\(\displaystyle y(t) = \begin{pmatrix}\cos(t)\\ \sin(t)\end{pmatrix}\)
 

sav26

New member
Joined
Nov 21, 2020
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3
Thanks for your response, but i can't really declare that \(\displaystyle ||y(t)||^3=1 \) (':
 

anon_hedgepig

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Nov 23, 2020
Messages
19
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.
 

sav26

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Nov 21, 2020
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3
Thanks! I managed to solve it (:)
 

anon_hedgepig

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Nov 23, 2020
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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.
 
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