General solution for system of differential equations

[Locoism]

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*e^λ2*t*v1 + b*e^λ2*t*v2 + c*e^λ3*t*v3 ?
I don't understand why, or what this means.

 

[Mark44]

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*e^λ2*t*v1 + b*e^λ2*t*v2 + c*e^λ3*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:

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

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

Since Y = PZ, then our solution is
$$Y = Y(t) = PZ = P\begin{bmatrix}e^{\lambda_1 t} & 0 & 0 \\ 0 & e^{\lambda_2 t} & 0 \\ 0 & 0 & e^{\lambda_3 t}\end{bmatrix}\begin{bmatrix}c_1 \\ c_2 \\ c_3 \end{bmatrix}$$

 

[Locoism]

OK thank you, that makes sense! But how come the diagonal matrix's entries are in the form e^tλ? And the matrix Z is given here by [c1, c2, c3]?

 

[Mark44]

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.

 

[Locoism]

Of course! Thank you, you are beautiful.

 

[Ray Vickson]

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*e^λ2*t*v1 + b*e^λ2*t*v2 + c*e^λ3*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