General solution for system of differential equations

In summary, the student is trying to solve a system of three differential equations. The eigenvalues are 5, -2, and 1, and the eigenvectors are [1, 0, 1], [0, 1, 0], and [1, 1, 0]. The system can be represented as a matrix equation, and if a diagonal matrix D is found that is similar to A, the system will be uncoupled. Once D is found, the student can solve for Y (the solution for Z is given by Z(t) = \begin{bmatrix}e^{\lambda_1 t} & 0 & 0 \\ 0 & e^{\lambda_2 t} & 0 \\
  • #1
81
0

Homework Statement


Find the general solution to the following differential equations
y'1 = -12y1 + 13y2 +10y3
y'2 = 4y1 - 3y2 - 4y3
y'3 = -21y1 +21y2 +19y3


The Attempt at a Solution


I'm a little unsure about what to do at the end, or what form to put it in.
The eigenvalues are
λ1 = 5, λ2 = -2, λ3 = 1
and the eigenvectors are
v1 = [1, -1, 3], v2 = [1, 0, 1], v3 = [1, 1, 0] (respectively)
so once I have that, what do I do? I need to put them into an exponential form like
a*eλ2*t*v1 + b*eλ2*t*v2 + c*eλ3*t*v3 ?
I don't understand why, or what this means.
 
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  • #2
Locoism said:

Homework Statement


Find the general solution to the following differential equations
y'1 = -12y1 + 13y2 +10y3
y'2 = 4y1 - 3y2 - 4y3
y'3 = -21y1 +21y2 +19y3


The Attempt at a Solution


I'm a little unsure about what to do at the end, or what form to put it in.
The eigenvalues are
λ1 = 5, λ2 = -2, λ3 = 1
and the eigenvectors are
v1 = [1, -1, 3], v2 = [1, 0, 1], v3 = [1, 1, 0] (respectively)
so once I have that, what do I do? I need to put them into an exponential form like
a*eλ2*t*v1 + b*eλ2*t*v2 + c*eλ3*t*v3 ?
I don't understand why, or what this means.

Your system of diff. equations is coupled, meaning that the three derivative components all involve the three y components. It's much easier to solve an uncoupled system such as
z1' = a1z1
z2' = a2z2
z3' = a3z3

and this is the primary motivation for diagonalizing a system of diff. equations.

Your system can be represented as a matrix equation as:
Y' = AY

If we can find a diagonal matrix D that is similar to A, then we can find a related system of equations that is uncoupled, so easier to solve.

Since you have already found that there are three distinct eigenvalues, that means your matrix A is diagonalizable. You have also found three eigenvectors for the three eigenvalues.

Form a matrix P whose columns are the eigenvectors. The order doesn't matter, but the order will determine which values appear on the diagonal of matrix D (the diagonal matrix).

Let Z = P-1Y, or equivalently, Y = PZ.

Then Z' = P-1Y' = P-1AY = P-1APZ = DZ, where D is the diagonal matrix. As already noted, this system is easy to solve, and its solution is given by:

[tex]Z = Z(t) = \begin{bmatrix}e^{\lambda_1 t} & 0 & 0 \\ 0 & e^{\lambda_2 t} & 0 \\ 0&0&e^{\lambda_3 t} \end{bmatrix} \begin{bmatrix}c_1 \\ c_2 \\ c_3 \end{bmatrix}[/tex]

We're not quite done. We're interested in Y = Y(t), not Z(t).

Since Y = PZ, then our solution is
[tex]Y = Y(t) = PZ = P\begin{bmatrix}e^{\lambda_1 t} & 0 & 0 \\ 0 & e^{\lambda_2 t} & 0 \\ 0 & 0 & e^{\lambda_3 t}\end{bmatrix}\begin{bmatrix}c_1 \\ c_2 \\ c_3 \end{bmatrix}[/tex]
 
  • #3
OK thank you, that makes sense! But how come the diagonal matrix's entries are in the form e? And the matrix Z is given here by [c1, c2, c3]?
 
  • #4
What's the general solution of z1' = λ1z1? Once you answer that, you might see that the matrix equation summarizes the solution for the vector Z.
 
  • #5
Of course! Thank you, you are beautiful.
 
  • #6
Locoism said:

Homework Statement


Find the general solution to the following differential equations
y'1 = -12y1 + 13y2 +10y3
y'2 = 4y1 - 3y2 - 4y3
y'3 = -21y1 +21y2 +19y3


The Attempt at a Solution


I'm a little unsure about what to do at the end, or what form to put it in.
The eigenvalues are
λ1 = 5, λ2 = -2, λ3 = 1
and the eigenvectors are
v1 = [1, -1, 3], v2 = [1, 0, 1], v3 = [1, 1, 0] (respectively)
so once I have that, what do I do? I need to put them into an exponential form like
a*eλ2*t*v1 + b*eλ2*t*v2 + c*eλ3*t*v3 ?
I don't understand why, or what this means.

If you think of a solution of the form y1 = a*exp(r*x), y2 = b*exp(r*x), y3 = c*exp(r*x) [same r in all three] you will get a cubic equation in r --- essentially, the characteristic equation in the matrix representation. This cubic has 3 roots (r1 = -2, r2 = 1, r3 = 5 in this case), and so your solution will have the form y1 = a1*exp(r1*x) + a2*exp(r2*x) + a3*exp(r3*x), etc. You get equations for the ai, bi and ci by substituting these expressions into the DE; the solutions are, essentially, constants times eigenvectors of the matrix. Finally, initial conditions serve to identify the constants.

RGV
 

1. What is a general solution for a system of differential equations?

A general solution for a system of differential equations is an equation or set of equations that describes the relationship between the dependent and independent variables in a system. It is a solution that includes all possible solutions to the system and can be used to find specific solutions for different initial conditions.

2. How is a general solution different from a particular solution?

A general solution describes all possible solutions to a system of differential equations, while a particular solution is a specific solution that satisfies both the equations and any given initial conditions. A particular solution can be found by substituting the initial conditions into the general solution.

3. Can a system of differential equations have more than one general solution?

Yes, a system of differential equations can have infinitely many general solutions. This is because there are many possible combinations of equations and constants that can be used to describe the relationship between the variables.

4. How can a general solution be verified?

A general solution can be verified by substituting it into the original system of differential equations and checking if it satisfies all the equations. It can also be verified by comparing it to other known solutions or by using numerical methods to solve the system.

5. Is a general solution always unique?

No, a general solution is not always unique. In some cases, there may be multiple general solutions that describe the same system of differential equations. This is because there may be different ways to represent the relationship between the variables in the system.

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