# 29.1 give a system of fundamental solutions

• MHB
• karush
In summary, to determine the general solution and give a system of fundamental solutions for a system of differential equations, one can use the techniques of diagonal, diagonalizable, or triangular matrices. The matrix A can be set as a 2x2 matrix and the eigenvalues can be found. A diagonal matrix has the property of A=PDP^-1 and a triangular matrix is one where the off-diagonal elements are zero. When the dependent variables do not appear in each other's equations, the system is called decoupled and can often be solved more easily.
karush
Gold Member
MHB
determine their general solution and give a system of fundamental solutions.
use the different techniques of diagonal, diagonalizedable, or triangular.
$\begin{cases} y'_1 & =3y_1 \\ y'_2 & =2y_2\end{cases}$
set matrix
$A= \begin{pmatrix}0 &3\\0 &2\end{pmatrix}$
then find eigenvaluesok just seeing if i am starting out correctly with this..
not sure what the difference is between diagonal and diagonalizedable? but it looks diagonal

karush said:
determine their general solution and give a system of fundamental solutions.
use the different techniques of diagonal, diagonalizedable, or triangular.
$\begin{cases} y'_1 & =3y_1 \\ y'_2 & =2y_2\end{cases}$
set matrix
$A= \begin{pmatrix}0 &3\\0 &2\end{pmatrix}$
then find eigenvaluesok just seeing if i am starting out correctly with this..
not sure what the difference is between diagonal and diagonalizedable? but it looks diagonal
It doesn't look diagonal to me, because of the $3$ that is off the diagonal. But it does look triangular.

Opalg said:
It doesn't look diagonal to me, because of the $3$ that is off the diagonal. But it does look triangular.

ok I don't know for sure but doesn't a diagonal have the property
 $A=PDP^{-1}$

which I think it does

Your DE doesn't match the matrix $A.$ Your system of DE's is actually de-coupled, as you've written it; for such a system, I would expect $A$ to be diagonal. So the matrix $A$ which matches your system would be
$$A=\left[\begin{matrix}3 &0 \\ 0 &2\end{matrix}\right].$$

how would that fit to $y_1$ and $y_2$ since there is no $x_i$

not sure what you mean by decoupled

A system of equations like yours:
\begin{align*}
y_1'&=3y_1 \\
y_2'&=2y_2
\end{align*}
is very often written (with an eye towards using matrix methods) in the form $\mathbf{y}'=A\mathbf{y},$ where $A$ is a $2\times 2$ matrix, and
$$\mathbf{y}=\left[\begin{matrix}y_1\\y_2\end{matrix}\right].$$
So, if you compare $\mathbf{y}'=A\mathbf{y}$ with your system, it appears that $A$ must be
$$A=\left[\begin{matrix}3&0\\0&2\end{matrix}\right].$$
The lack of $x_i$ is not a problem here.

The system is called "decoupled" when the dependent variables don't show up in each others' DE's. So, in the equation $y_1'=3y_1,$ there's no $y_2,$ and in the equation $y_2'=2y_2,$ there's no $y_1.$ De-coupled DE's are very often considerably simpler to solve, since you can basically solve each one separately. Indeed, you can simply write down by inspection the solution to this system:
\begin{align*}
y_1&=C_1 e^{3x}\\
y_2&=C_2 e^{2x},
\end{align*}
assuming $x$ is the independent variable.

## 1. What is the purpose of a system of fundamental solutions?

A system of fundamental solutions is used to represent a general solution to a linear differential equation. It helps to simplify the process of finding solutions to a differential equation by providing a set of functions that can be combined to form a solution.

## 2. How is a system of fundamental solutions different from a general solution?

A general solution is a single equation that encompasses all possible solutions to a differential equation. A system of fundamental solutions, on the other hand, is a set of equations that can be combined to form a general solution.

## 3. How do you find a system of fundamental solutions?

To find a system of fundamental solutions, you can use a method called the method of undetermined coefficients. This involves assuming a general form for the solutions and then solving for the coefficients using the differential equation.

## 4. Can a system of fundamental solutions be used for all types of differential equations?

No, a system of fundamental solutions is only applicable to linear differential equations. Non-linear differential equations require different methods for finding solutions.

## 5. What is the advantage of using a system of fundamental solutions?

The advantage of using a system of fundamental solutions is that it simplifies the process of finding solutions to a differential equation. It provides a set of functions that can be easily combined to form a general solution, rather than having to solve a complex equation each time.

Replies
4
Views
1K
Replies
2
Views
2K
Replies
2
Views
2K
Replies
4
Views
2K
Replies
2
Views
2K
Replies
4
Views
2K
Replies
34
Views
2K
Replies
1
Views
3K
Replies
7
Views
2K
Replies
3
Views
2K