# Write an Equation given a solution to an ODE

1. Oct 13, 2015

### joshmccraney

1. The problem statement, all variables and given/known data
Give an example of a system of differential equations for which $(t,1)$ is a solution.

2. Relevant equations
Nothing comes to mind.

3. The attempt at a solution
I thought to initial pose the system as an eigenvalue problem $\vec{x}' = A \vec{x}$. However, $(t,1)$ is generally not an eigenvalue. Any ideas?

Would any system work you think, not necessarily $\vec{x}' = A \vec{x}$?

Thanks!

2. Oct 13, 2015

### Staff: Mentor

$\frac{dx}{dt} = t$
$\frac{dy}{dt} = 1$
Can you solve for x and y?

3. Oct 13, 2015

### LCKurtz

Two questions:
1. Can you come up with a second order linear DE with those two solutions?
2. Do you know how to write a second order as a 2 by 2 matrix system?

4. Oct 15, 2015

### pasmith

Rather than guessing, it is easiest to work backwards from the required solution until you have an appropriate system. Here you just need "a system of differential equations" (with no additional restrictions on the type of system) which admits the required solution, and differentiating the required solution once will get you "a system of differential equations" which admits the required solution.

That said, there is a system of the form $\vec{x}' = A \vec {x}$; since $1 = e^{0t}$ and $t = te^{0t}$ you are looking for a matrix which has two zero eigenvalues but is not the zero matrix.