Question

4. Solve the following initial-value problem: 2 2 for 3-dimensional vector X. Present the final answer in terms of єkt, ektsi

0 0
Add a comment Improve this question Transcribed image text
Answer #1

Solution:

According to the matrix we have three equations

x''=4x, x(0)=5,x'(0)=2

y''=3y-4z, y(0)=2,y'(0)=4

z''=4y+3z, z(0)=0,z'(0)=2

the first one is easy and the solution is, r(t)32 +2e21

for the second two we apply laplace transform to both sides

s^2Y(s)-sy(0)-y'(0)=3Y(s)-4Z(s)=>\left (s^2-3 \right )Y(s)=2s+4-4Z(s)

s^2Z(s)-sz(0)-z'(0)=4Y(s)+3Z(s)=>\left (s^2-3 \right )Z(s)=2+4Y(s)

eliminating Z(s) we get,

=>\left (s^2-3 \right )Y(s)=2s+4-4\left (\frac{2+4Y(s)}{\left (s^2-3 \right )} \right )

=>\left (s^2-3 \right )Y(s)=2s+4-\frac{8}{\left (s^2-3 \right )}-\frac{16Y(s)}{\left (s^2-3 \right )}

=>\left (s^2-3 \right )Y(s)+\frac{16Y(s)}{\left (s^2-3 \right )}=2s+4-\frac{8}{\left (s^2-3 \right )}

2162s+ 2 3 (s2 - 3)

=>Y(s)=\frac{\left (2s+4 \right )\left (s^2-3 \right )}{\left (s^2-3 \right )^2+16}-\frac{8}{\left (s^2-3 \right )^2+16}

=>Y(s)=2\left (\frac{s-2}{s^2-4s+5} \right )

=>Y(s)=2\left (\frac{s-2}{\left ( s-2 \right )^2+1} \right )

taking inverse laplace transform we get,

=>y(t)=2e^{2t}\cos (t)

\text{Now, }

=>Z(s)=\frac{2}{\left (s^2-3 \right )}+8\left (\frac{s-2}{\left (s^2-4s+5 \right )\left ( s^2-3 \right )} \right )

=>Z(s)=\frac{2\left (s^2-4s+5 \right )+8s-16}{\left (s^2-4s+5 \right )\left ( s^2-3 \right )}

=>Z(s)=\frac{2s^2-6}{\left (s^2-4s+5 \right )\left ( s^2-3 \right )}

=>Z(s)=\frac{1}{\left (s^2-4s+5 \right )}

=>Z(s)=\frac{1}{\left (s-2 \right )^2+1}

taking inverse laplace transform we get,

z(t)et sin(t

\text{Therefore, the three solutions are }

=>x(t)=3e^{2t}+2e^{-2t},y(t)=e^{2t}\cos (t),z(t)=e^{2t}\sin (t)

Add a comment
Know the answer?
Add Answer to:
4. Solve the following initial-value problem: 2 2 for 3-dimensional vector X. Present the final answer in terms of єkt, ektsinrnt and ekt cos mt. 4. Solve the following initial-value problem: 2...
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for? Ask your own homework help question. Our experts will answer your question WITHIN MINUTES for Free.
Similar Homework Help Questions
ADVERTISEMENT
Free Homework Help App
Download From Google Play
Scan Your Homework
to Get Instant Free Answers
Need Online Homework Help?
Ask a Question
Get Answers For Free
Most questions answered within 3 hours.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT