Construct a differential equation from the basis of solution

In summary: Thus in summary, to construct the desired matrix A, we can use the given solutions to determine the images of (1,0,0), (0,1,0), and (0,0,1) and put them together as the columns of A.
  • #1
Mr Davis 97
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Homework Statement


Write down a 3x3 matrix A such that the equation ##\vec{y}'(t) = A \vec{y}(t)## has a basis of solutions ##y_1=(e^{-t},0,0),~~y_2 = (0,e^{2t},e^{2t}),~~y_3 = (0,1,-1)##

Homework Equations

The Attempt at a Solution


I was thinking that, it looks like the matrix would have to have eigenvalues -1, 2, 0, with corresponding eignevectors (1,0,0), (0,1,1), and (0,1,-1). However, I am not sure how I would use this information to construct a matrix.
 
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  • #2
Mr Davis 97 said:

Homework Statement


Write down a 3x3 matrix A such that the equation ##\vec{y}'(t) = A \vec{y}(t)## has a basis of solutions ##y_1=(e^{-t},0,0),~~y_2 = (0,e^{2t},e^{2t}),~~y_3(0,1,-1)##

Homework Equations

The Attempt at a Solution


I was thinking that, it looks like the matrix would have to have eigenvalues -1, 2, 0, with corresponding eignevectors (1,0,0), (0,1,1), and (0,1,-1). However, I am not sure how I would use this information to construct a matrix.

Are you sure that the third basis member is ##y_3 = (0,1,-1)?## I ask because ##y_3\prime \equiv (0,0,0)## and that is not a non-zero linear combination of ##y_1, y_2, y_3##, which means that the problem looks impossible.
 
  • #3
Ray Vickson said:
Are you sure that the third basis member is ##y_3 = (0,1,-1)?## I ask because ##y_3\prime \equiv (0,0,0)## and that is not a non-zero linear combination of ##y_1, y_2, y_3##, which means that the problem looks impossible.
Yes, I am sure. One answer to the question that the author gives is ##\begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 1 \end{bmatrix}##, if that helps.
 
  • #4
If we look at the solutions, we'll have again two blocks here: the first coordinate with the solution ##\vec{y}_1## and the other two with ##\vec{y}_2,\vec{y}_3##. Let's write ##\vec{y}(t)=(x_1(t),x_2(t),x_3(t))## to not confuse the coordinates of ##\vec{y}(t)## with the numbering of the solutions.

So we have in the first coordinate the equation ##x_1'(t)=a_{11}x_1(t)## which is solved by ##x_1(t)=\exp(a_{11}t)+C##. Since ##x_1(t)=(\vec{y}_1(t))_1=\exp(-t)## is a given solution, we can chose ##a_{11}=-1## and ##C=0##.

Thus we are left with ##\vec{y}'(t)=\begin{bmatrix}-1&0\\0&A\end{bmatrix}\cdot \vec{y}(t)## or ##\begin{bmatrix}x_2'(t)\\x_3'(t)\end{bmatrix}=A\cdot \begin{bmatrix}x_2(t)\\x_3(t)\end{bmatrix} = \begin{bmatrix}a&b\\c&d\end{bmatrix}\cdot \begin{bmatrix}x_2(t)\\x_3(t)\end{bmatrix}##.

Now we have to solve these two equations, compare the solutions with ##y_2(t)=(e^{2t},e^{2t})## and ##y_3(t)=(1,-1)## to determine ##a,b,c,d## the way we did with ##a_{11}=-1## in the first coordinate.
(I haven't done it. It's just how I would approach the problem.)
 
  • #5
Mr Davis 97 said:

Homework Statement


Write down a 3x3 matrix A such that the equation ##\vec{y}'(t) = A \vec{y}(t)## has a basis of solutions ##y_1=(e^{-t},0,0),~~y_2 = (0,e^{2t},e^{2t}),~~y_3 = (0,1,-1)##

Homework Equations

The Attempt at a Solution


I was thinking that, it looks like the matrix would have to have eigenvalues -1, 2, 0, with corresponding eignevectors (1,0,0), (0,1,1), and (0,1,-1). However, I am not sure how I would use this information to construct a matrix.

The first column of the matrix is the image of (1,0,0). The second column is the image of (0,1,0). The third column is the image of (0,0,1).

Now you already know the image of (1,0,0), and from the known images of (0,1,1) and (0,1,-1) you can find the images of (0,1,0) and (0,0,1).
 

Related to Construct a differential equation from the basis of solution

1. What is a differential equation?

A differential equation is a mathematical equation that relates an unknown function to its derivatives. It describes the rate of change of a system over time, and it is often used in modeling real-world phenomena in fields such as physics, engineering, and economics.

2. How do you construct a differential equation from a basis of solution?

To construct a differential equation from a basis of solution, you first need to determine the general solution of the differential equation. This can be done by using known techniques such as separation of variables or integrating factors. Then, you can substitute your basis of solution, which is a specific solution of the differential equation, into the general solution and solve for the unknown parameters to obtain the final differential equation.

3. What is the importance of constructing differential equations from basis of solutions?

Constructing differential equations from basis of solutions allows us to model and understand complex systems and phenomena in a more mathematical and precise way. It also allows us to predict future behavior of the system and make informed decisions based on the solutions obtained from the differential equations.

4. Can a differential equation have multiple basis of solutions?

Yes, a differential equation can have multiple basis of solutions. This means that there can be multiple specific solutions that satisfy the differential equation and can be substituted into the general solution to obtain the final differential equation.

5. Is there a standard method for constructing differential equations?

There is no single standard method for constructing differential equations. The method used depends on the type of differential equation and the known information about the system. Some common techniques include separation of variables, integrating factors, and variation of parameters.

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