How to check a particular solution of System of Linear ODEs?

In summary, to check if a given solution works for a system of inhomogeneous linear ODEs, one can use symbolic differentiation in Mathematica to verify that the proposed solution satisfies the ODEs.
  • #1
351
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If I have been given a system of inhomogeneous linear ODEs,
$$
\vec{x'} =
\begin{bmatrix}
4 & -1 \\
5 & -2 \\
\end{bmatrix}
\vec{x}
+
\begin{bmatrix}
18e^{2t} \\
30e^{2t}\\
\end{bmatrix}
$$

I have found its particular solution to be:
$$
1/4
\begin{bmatrix}
-31e^{2t} - 25e^{6t} \\
85e^{2t} - 25e^{6t} \\
\end{bmatrix}
$$

But this answer doesn't match with the answer given in the book. Can someone tell me how to check if this solutions works by writing some code in Mathematica? I know, I can use DSolve for solving them, but I'm asking a reverse of that.

Please guide me step by step, I'm new to Mathematica and I don't have any background in programming.
 
Last edited:
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  • #2
Hall said:
If I have been given a system of inhomogeneous linear ODEs,
$$
\vec{x'} =
\begin{bmatrix}
4 & -1 \\
5 & -2 \\
\end{bmatrix}
\vec{x}
+
\begin{bmatrix}
18e^{2t} \\
30e^{2t}\\
\end{bmatrix}
$$

I have found its particular solution to be:
$$
1/4
\begin{bmatrix}
-31e^{2t} - 25e^{6t} \\
85e^{2t} - 25e^{6t} \\
\end{bmatrix}
$$

But this answer doesn't match with the answer given in the book.

I look at your solution, and I must ask myself: Where does [itex]e^{6t}[/itex] come from? [itex]e^{2t}[/itex] is an eigenfunction of the derivative operator: [itex](e^{2t})' = 2e^{2t}[/itex]. So I would expect the particular function to be [itex]ae^{2t}[/itex] for some constant vector [itex]a[/itex], which can be determined by substituting this into the ODE. That your answer is not of this form, or of this form plus a complementary function (6 is not an eigenvalue of the matrix, so a multiple of [itex]e^{6t}[/itex] is not a complementary function), leads me to suspect that you have made an error, but since you haven't shown your working I can't tell you what it is.

Can someone tell me how to check if this solutions works by writing some code in Mathematica? I know, I can use DSolve for solving them, but I'm asking a reverse of that.

Please guide me step by step, I'm new to Mathematica and I don't have any background in programming.

The answer to this question is "symbolic differentiation". Define your proposed solution as a function, and check to see that
Code:
f'[t] - {{4, -1},{5, -2}} . f[t] - {18*exp[2*t], 30*exp[2*t]}
is zero. Have a look at the examples at https://reference.wolfram.com/language/ref/Derivative.html.
 
Last edited:
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Likes Hall

1. How can I check if a given solution is valid for a system of linear ODEs?

To check the validity of a solution for a system of linear ODEs, you can substitute the solution into the original equations and see if it satisfies them. If the solution satisfies all the equations, then it is a valid solution for the system of linear ODEs.

2. What is the process for verifying a solution to a system of linear ODEs?

The process for verifying a solution to a system of linear ODEs involves substituting the solution into each equation and checking if it satisfies the equation. If the solution satisfies all the equations, then it is a valid solution for the system of linear ODEs.

3. Can I use a numerical method to check a solution for a system of linear ODEs?

Yes, you can use a numerical method such as Euler's method or Runge-Kutta method to approximate the solution of a system of linear ODEs and then check if the approximation satisfies the original equations. However, this method may not always be accurate, so it is recommended to also check the solution analytically if possible.

4. How do I know if my solution is the only solution to a system of linear ODEs?

In general, a system of linear ODEs can have multiple solutions. To determine if your solution is the only one, you can solve the system of equations using different initial conditions. If all the solutions obtained are the same, then your solution is the only solution for that particular system of linear ODEs.

5. Can I use a graph to check a solution for a system of linear ODEs?

Yes, you can plot the given solution as a function of the independent variable and see if it satisfies the original equations. However, this method may not always be accurate and it is recommended to also check the solution analytically if possible.

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