• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Diff. EQ. Inhomogeneous Linear System (Double check please)

  • Thread starter VinnyCee
  • Start date
489
0
Here is the problem:

[tex]X'\,=\,\left(\begin{array}{cc} 0 & 2 \\ -1 & 3 \end{array}\right)\,X\,+\left(\begin{array}{c} 2 \\ e^{-3t}\end{array}\right)[/tex]

Here is what I have:

[tex]r_1\,=\,2,\,\,\,\,\,\,r_2\,=\,1[/tex]

[tex]u_1'\,=\,-2\,e^{-2t}\,+2\,e^{-5t}[/tex]

[tex]u_2'\,=\,2\,e^{-t}\,-\,e^{-4t}[/tex]

[tex]\psi\,=\,\left(\begin{array}{cc} e^{2t} & 2\,e^t \\ e^{2\,t} & e^t \end{array}\right)[/tex]

After integrating the [itex]u_1[/itex] and [itex]u_2[/itex] terms and multiplying the answer by [itex]\psi[/itex], I get:

[tex]X\,=\,\left(\begin{array}{c} C_1\,e^{2t}\,+\,2\,C_2\,e^t\,+\,\frac{1}{10}\,e^{-3t}\,-\,3 \\ C_1\,e^{2t}\,+\,C_2\,e^t\,-\,\frac{3}{20}\,e^{-3t}\,-\,1 \end{array}\right)[/tex]

Then seperating and combining terms:

[tex]X\,=\,C_1\,e^{2t}\,\left(\begin{array}{c} 1 \\ 1 \end{array}\right)\,+\,C_2\,e^t\,\left(\begin{array}{c} 2 \\ 1 \end{array}\right)\,+\,\frac{1}{20}\,\left(\begin{array}{cc} 2\,e^{-3t}\,-\,60 \\ 3\,e^{-3t}\,-\,20 \end{array}\right)[/tex]

Does this look correct? Thanks.
 
489
0
Any ideas?

Anyone have the same thing when solving this problem?
 

saltydog

Science Advisor
Homework Helper
1,582
2
VinnyCee said:
Anyone have the same thing when solving this problem?
Hello VinnyCee. Takes a while. It's a challenging problem. Your answer is NOT correct: Missing a minus sign for the second equation. You know, you can back-substitute the answer into the original ODE to verify the answer or if you're lazy like me, just input the results into Mathematica and have it back-substitute and compare the results. :smile:

I get:

[tex]x_p(t)=-3+1/10 e^{-3t}[/tex]

[tex]y_p(t)=-1-3/20e^{-3t}[/tex]

Hope you didn't get my first (unedited) post where I said it was correct. Didn't notice the minus sign missing. :surprised
 
Last edited:
489
0
Could you show me that?

This backsubstitution that you mention. How do you type this? Can you show me the routine you did in Mathematica, so I can double check others also?
 

saltydog

Science Advisor
Homework Helper
1,582
2
VinnyCee said:
This backsubstitution that you mention. How do you type this? Can you show me the routine you did in Mathematica, so I can double check others also?
Code:
x[t_]:=-3+1/10 e^(-3 t);
y[t_]:=-1-3/20 e^(-3 t);
D[x[t],t]
Simplify[2 y[t]+2]
D[y[t],t]
Simplify[-x[t]+3 y[t]+e^(-3 t)
Just define the solutions (x[t] and y[t]), calculate the derivatives, then insert the x[t] and y[t] into the right hand side and compare the resuts.
 

Related Threads for: Diff. EQ. Inhomogeneous Linear System (Double check please)

Replies
16
Views
2K
Replies
1
Views
414
Replies
3
Views
1K
Replies
9
Views
1K
Replies
3
Views
4K
  • Posted
Replies
5
Views
1K
Replies
2
Views
2K
  • Posted
Replies
1
Views
3K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top