A system of 1st order nonlinear differential equations

[quacam09]

Hello,

Can you give some suggestions to solve the following system of 1st order nonlinear differential equations?

Thank you.

$$\[ \begin{array}{l} u'(t) = Au^2 (t) + B(t)u + C(t) \\ u(t) = \left[ {\begin{array}{*{20}c} {x_1 (t)} \\ {x_2 (t)} \\ \end{array}} \right] \\ A = \left[ {\begin{array}{*{20}c} {a_{11} } & {a_{12} } \\ {a_{21} } & {a_{22} } \\ \end{array}} \right] \\ B(t) = \left[ {\begin{array}{*{20}c} {f_{11} (t)} & {f_{12} (t)} \\ {f_{21} (t)} & {f_{22} (t)} \\ \end{array}} \right] \\ C(t) = \left[ {\begin{array}{*{20}c} {g_{11} (t)} & {g_{12} (t)} \\ {g_{21} (t)} & {g_{22} (t)} \\ \end{array}} \right] \\ \end{array} \]$$

 

[Mark44]

Hello,

Can you give some suggestions to solve the following system of 1st order nonlinear differential equations?

Thank you.

$$\[ \begin{array}{l} u'(t) = Au^2 (t) + B(t)u + C(t) \\ u(t) = \left[ {\begin{array}{*{20}c} {x_1 (t)} \\ {x_2 (t)} \\ \end{array}} \right] \\ A = \left[ {\begin{array}{*{20}c} {a_{11} } & {a_{12} } \\ {a_{21} } & {a_{22} } \\ \end{array}} \right] \\ B(t) = \left[ {\begin{array}{*{20}c} {f_{11} (t)} & {f_{12} (t)} \\ {f_{21} (t)} & {f_{22} (t)} \\ \end{array}} \right] \\ C(t) = \left[ {\begin{array}{*{20}c} {g_{11} (t)} & {g_{12} (t)} \\ {g_{21} (t)} & {g_{22} (t)} \\ \end{array}} \right] \\ \end{array} \]$$

What does u²(t) mean? Is it u(t) $\cdot$ u(t)?

Also, shouldn't the differential equation be
$$u'(t) = Au^2 (t) + B(t)u(t) + C(t)$$
?

 

[HallsofIvy]

Hello,

Can you give some suggestions to solve the following system of 1st order nonlinear differential equations?

Thank you.

[tex]

\[
\begin{array}{l}
u'(t) = Au^2 (t) + B(t)u + C(t) \\

As Mark44 notes, the "u^2" doesn't make sense here. If it is the dot product, then multiplying it by a two by two matrix doesn't make sense. If it is the cross product, then Both Au^2 and Bu are 2 dimensional vectors but you cannot add that to C(t), a two by two matrix.

u(t) = \left[ {\begin{array}{*{20}c}
{x_1 (t)} \\
{x_2 (t)} \\
\end{array}} \right] \\
A = \left[ {\begin{array}{*{20}c}
{a_{11} } & {a_{12} } \\
{a_{21} } & {a_{22} } \\
\end{array}} \right] \\
B(t) = \left[ {\begin{array}{*{20}c}
{f_{11} (t)} & {f_{12} (t)} \\
{f_{21} (t)} & {f_{22} (t)} \\
\end{array}} \right] \\
C(t) = \left[ {\begin{array}{*{20}c}
{g_{11} (t)} & {g_{12} (t)} \\
{g_{21} (t)} & {g_{22} (t)} \\
\end{array}} \right] \\
\end{array}
\]

[/tex]

 

[EnumaElish]

For a numeric solution the Runge-Kutta method seems an especially good method to use: http://www.springerlink.com/content/w080u7262137j867/

 

[Mark44]

For some reason, HallsOfIvy's reply didn't render correctly. Here it is.

As Mark44 notes, the "u^2" doesn't make sense here. If it is the dot product, then multiplying it by a two by two matrix doesn't make sense. If it is the cross product, then Both Au^2 and Bu are 2 dimensional vectors but you cannot add that to C(t), a two by two matrix.

 

[quacam09]

For some reason, HallsOfIvy's reply didn't render correctly. Here it is
As Mark44 notes, the "u^2" doesn't make sense here. If it is the dot product, then multiplying it by a two by two matrix doesn't make sense. If it is the cross product, then Both Au^2 and Bu are 2 dimensional vectors but you cannot add that to C(t), a two by two matrix..

Thanks for your help and sorry for unclear things.

u^2 is a cross product. It means
$$\[ u^2 (t) = \left[ {\begin{array}{*{20}c} {x_1^2 (t)} \\ {x_2^2 (t)} \\ \end{array}} \right] \\ \]$$

And C(t)
$$C(t) = \left[ {\begin{array}{*{20}c} {g_{1} (t)} \\ {g_{2} (t)} \\ \end{array}} \right] \\$$

 

[quacam09]

For a numeric solution the Runge-Kutta method seems an especially good method to use: http://www.springerlink.com/content/w080u7262137j867/

Thanks EnumaElish. Is an analytical solution impossible? If there is a method to obtain an analytical solution, can you suggest me?