# Second-Order Nonlinear Differential Equation

#### sav26

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

#### anon_hedgepig

##### New member
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

##### New member
Thanks! I managed to solve it (

#### anon_hedgepig

##### New member
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.