# A system of 1st order nonlinear differential equations

1. Jun 18, 2010

### quacam09

Hello,

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

Thank you.

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

2. Jun 18, 2010

### Staff: Mentor

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

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

3. Jun 18, 2010

### HallsofIvy

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.

4. Jun 18, 2010

### EnumaElish

5. Jun 18, 2010

### Staff: Mentor

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

6. Jun 23, 2010

### quacam09

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

7. Jun 23, 2010

### quacam09

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