System of ODEs...challenge! (dM(s)/ds) = D * w(s), (dw(s)/ds) = w(s) * M(s) * (1/s^2)

[Dc89]

Hi guys,

I've been having a hard time solving this system of ODEs:

[math]\frac{dM(s)}{ds}=D\cdot w(s)[/math][math]\frac{dw(s)}{ds}=w(s)\cdot M(s)\cdot \frac{1}{s^2}[/math]
where D is a positive constant, s>0 and the boundary conditions are:

[math]M(s_1)=\sigma M(s_2)=\sigma_2[/math]
I've already tried transforming it into a 2nd order diff. equation, but it doesn't work because M''(s) becomes a function of w(s) and I can't solve it. It's also not possible to express it in matrix notation to find eigenvalues and eigenvectors since w(s)*M(s) are stuck together in the second equation. My last hope was to divide one equation by the other and try to solve the ODE dw/dM, but also ended up with bad results...

can someone help me?

 

[Dc89]

It's also important to point out that the range of this function M(s) is [math]\Sigma={(\sigma_k \in \mathbb{R} \mid \sigma \leq \sigma_k \leq \sigma_2)}[/math] while the domain is the set [math]S={(s \in \mathbb{R}_{>0} \mid s_1 \leq s \leq s_2)}[/math]

 

[topsquark]

It's also important to point out that the range of this function M(s) is [math]\Sigma={(\sigma_k \in \mathbb{R} \mid \sigma \leq \sigma_k \leq \sigma_2)}[/math] while the domain is the set [math]S={(s \in \mathbb{R}_{>0} \mid s_1 \leq s \leq s_2)}[/math]

Solve the first equation for w(s) and put it into the second equation. That will give you a second order equation for M(s).

-Dan

 

[khansaheb]

Hi guys,

I've been having a hard time solving this system of ODEs:

[math]\frac{dM(s)}{ds}=D\cdot w(s)[/math][math]\frac{dw(s)}{ds}=w(s)\cdot M(s)\cdot \frac{1}{s^2}[/math]
where D is a positive constant, s>0 and the boundary conditions are:

[math]M(s_1)=\sigma M(s_2)=\sigma_2[/math]
I've already tried transforming it into a 2nd order diff. equation, but it doesn't work because M''(s) becomes a function of w(s) and I can't solve it. It's also not possible to express it in matrix notation to find eigenvalues and eigenvectors since w(s)*M(s) are stuck together in the second equation. My last hope was to divide one equation by the other and try to solve the ODE dw/dM, but also ended up with bad results...

can someone help me?

\(\displaystyle \frac{dM(S)}{ds} = D * w(s)\) ...........differentiate again

\(\displaystyle \frac{d^2}{ds^2}M(S) = D * \frac{d}{ds}w(s) = D * w(s) * M(s) * (1/s^2) = D * M(s) * (1/s^2) * (1/D) * \frac{d}{ds}M(s)\) .... continue