# -16.1 Find the general solution to the system of DE

• MHB
• karush
In summary, the solution to the system of differential equations is $y= Ae^{3x}$, where $A$ and $B$ are constants.
karush
Gold Member
MHB
Find the general solution to the system of differential equations
$\begin{cases} y'_1&=2y_1+y_2-y_3 \\ y'_2&=3y_2+y_3\\ y'_3&=3y_3 \end{cases}$
let
$y(t)=\begin{bmatrix}{y_1(t)\\y_2(t)\\y_3(t)}\end{bmatrix} ,\quad A=\begin{bmatrix} 2 & 1 & -1 \\ 0 & 3 & 1 \\ 0 & 0 & 3 \end{bmatrix}$
so
\begin{align*}\displaystyle
y_1&=c_1e^{2t}+c_2e^{t}+c_3e^{-t}\\
y_2&=c_2e^{3t}+c_3e^{t}\\
y_3&=c_3e^{3t}
\end{align*}

ok if correct so far assume next step is to diagonalize $A:\quad A=PDP^{-1}$

well according to EMH this is not diagonalizable but is look like a triangle

so would this be

$\begin{pmatrix} y'_1 \\ y'_2 \\ y'_3 \end{pmatrix} = \begin{pmatrix} 2y_1&+y_2&-y_3 \\ 0&3y_2&y_3\\ 0&0&3y_3 \end{pmatrix} \cdot \begin{pmatrix} x \\ y \\ z \end{pmatrix} + \begin{pmatrix} b_1(t) \\ b_2(t) \\ b_3(t) \end{pmatrix}$

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What "next step"? The problem asked you to find y1, y2, and y3 and you have already done that!

(Unfortunately you also have the wrong solution!)

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Personally, I would not use "matrices" at all.

karush said:
Find the general solution to the system of differential equations
$\begin{cases} y'_1&=2y_1+y_2-y_3 \\ y'_2&=3y_2+y_3\\ y'_3&=3y_3 \end{cases}$
\
The last equation, $y'_3= 3y_3$ is a first order, linear, homogeneous equation with constant coefficients. It's "characteristic equation" is $r= 3$ so the general solution is $y= Ae^{3x}$. (You could also have written this as $\frac{dy}{y}= 3dx$ and integrate to get the same solution.)

Once you know that the second equation can be written as $y'_2= 3y_2+ Ae^{3x}$, another first order linear equation with constant coefficients but now it is not homogeneous. Its characteristic equation is again $r= 3$ and the general solution to the "associated homogeneous equation" is again $y= Be^{3x}$. With right side "$Ae^{3t}$" we would normally try a solution to the entire equation of the form "$e^{3x}$ but because that satisfies the homogenous equation we try, instead, $y= Pxe^{3x}$. Then $y'= Pe^{3x}+ 3Pxe^{3x}$ and $e^{3x}+ 3Pe^{3x}= 3Pxe^{3x}+ Ae^{3x}$ so $P= A$ and $y_2= Be^{3x}+ Axe^{3x}$.

Now we can write the first equation as
$y'_1= 2y_1+ A^{3x}- Be^{3x}- Axe^{3x}= 2y_1+ (A- B)e^{3x}- Axe^{3x}$ and can be solved in much the same way.

## 1. What is a system of DE?

A system of DE (differential equations) is a set of equations that involve derivatives of one or more unknown functions. These equations are used to model real-world phenomena in fields such as physics, engineering, and economics.

## 2. What is the general solution to a system of DE?

The general solution to a system of DE is a set of equations that satisfies all of the equations in the system. It includes all possible solutions, and may contain arbitrary constants that can be determined by applying initial conditions.

## 3. How do you find the general solution to a system of DE?

To find the general solution, you must first solve each equation in the system separately. Then, you can combine these solutions to form the general solution by including any arbitrary constants that arise from the individual solutions.

## 4. What is the role of initial conditions in finding the general solution?

Initial conditions are specific values of the unknown functions and their derivatives at a given point. They are used to determine the values of the arbitrary constants in the general solution, thus providing a unique solution to the system of DE.

## 5. Can the general solution to a system of DE be expressed in a closed form?

It depends on the system of DE. In some cases, the general solution can be expressed in a closed form using known mathematical functions. However, in more complex systems, the general solution may only be expressible as a series or an integral.

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