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.

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

 

[Mark44]

Hello,

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

Thank you.

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

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.

$$<br /> \[<br /> \begin{array}{l}<br /> 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.<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> u(t) = \left[ {\begin{array}{*{20}c}<br /> {x_1 (t)} \\<br /> {x_2 (t)} \\<br /> \end{array}} \right] \\ <br /> A = \left[ {\begin{array}{*{20}c}<br /> {a_{11} } & {a_{12} } \\<br /> {a_{21} } & {a_{22} } \\<br /> \end{array}} \right] \\ <br /> B(t) = \left[ {\begin{array}{*{20}c}<br /> {f_{11} (t)} & {f_{12} (t)} \\<br /> {f_{21} (t)} & {f_{22} (t)} \\<br /> \end{array}} \right] \\ <br /> C(t) = \left[ {\begin{array}{*{20}c}<br /> {g_{11} (t)} & {g_{12} (t)} \\<br /> {g_{21} (t)} & {g_{22} (t)} \\<br /> \end{array}} \right] \\ <br /> \end{array}<br /> \] </div> </div> </blockquote>$$

 

[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
$$<br /> <br /> \[<br /> u^2 (t) = \left[ {\begin{array}{*{20}c}<br /> {x_1^2 (t)} \\<br /> {x_2^2 (t)} \\<br /> \end{array}} \right] \\ <br /> <br /> \]$$

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

 

[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?