Complex-valued solutions to real-valued solutions

  • Thread starter NINHARDCOREFAN
  • Start date
In summary, Two eigen value solutions, r=-1+-2i, were given for a problem along with their corresponding eigen vectors. The goal is to convert these complex solutions to real-valued solutions. The book used the eigen vector (-2i, 1) to come up with the solution x = c * e^(-t)( -2sin(2t), cos(2t) ) + d * e^(-t)( 2cos(2t), sin(2t) ), where c and d are constants. It is known that the real value of r=1+-2i should be e^(-t)sin(2t) + e^(-t)cos(2t), but it
  • #1
NINHARDCOREFAN
118
0
I'm going crazy trying to figure out how to do this. Here are two eigen value solutions: r=-1+-2i to a problem. And the two eigen vectors are (-2i, 1) and (2i, 1). How do I covert this to a real-valued solution? The book used (-2i, 1) to come up with the solution x = c * e^(-t)( -2sin(2t), cos(2t) ) + d * e^(-t)( 2cos(2t), sin(2t) )
c and d are constants.
I know that real value of r=1+-2i should be e^(-t)sin(2t) + e^(-t)cos(2t) but I don't how they came up with their answer using this fact.
 
Physics news on Phys.org
  • #2
[tex]e^{ix}= cos(x)+i sin(x)[/tex]. Does that help?
 
Last edited by a moderator:
  • #3
[tex]e^{ix}= cos(x)+i sin(x)[/tex]
 

1. What is the difference between complex-valued solutions and real-valued solutions?

The main difference between complex-valued solutions and real-valued solutions is that complex-valued solutions involve the use of complex numbers, which have both a real and imaginary component, while real-valued solutions only involve real numbers. Complex-valued solutions are often used in math and science to solve problems that cannot be solved with real numbers alone.

2. Why are complex-valued solutions necessary for certain problems?

Complex-valued solutions are necessary for certain problems because they allow for a more complete and accurate representation of the problem. Some problems, such as those involving waves or oscillations, cannot be fully described using only real numbers. Complex numbers provide a way to represent both the magnitude and phase of these types of phenomena.

3. How do complex-valued solutions relate to real-world applications?

Complex-valued solutions have many real-world applications, particularly in fields such as engineering, physics, and economics. They are used to model and analyze systems that involve oscillations, such as electrical circuits, mechanical systems, and financial markets. Complex-valued solutions also play a key role in signal processing and communication systems.

4. Can complex-valued solutions have real-valued solutions?

Yes, it is possible for complex-valued solutions to have real-valued solutions. In fact, many complex-valued equations have both real and complex solutions. This is because the solutions to these equations can involve both real and imaginary numbers, depending on the specific values of the variables involved.

5. How are complex-valued solutions related to the imaginary unit?

The imaginary unit, denoted by the letter i, is a fundamental part of complex-valued solutions. It is defined as the square root of -1, and allows us to represent the imaginary component of a complex number. In complex-valued solutions, the imaginary unit is often used to represent the phase or angular component of a waveform or oscillation.

Similar threads

Linear homogenous system with repeated eigenvalues
    • Calculus and Beyond Homework Help
Replies
2
Views
507
Partial fractions with complex linear terms
    • Calculus and Beyond Homework Help
Replies
8
Views
142
Characterize Fourier coefficients
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Initial Value w/ Vector-Valued Function
    • Calculus and Beyond Homework Help
Replies
11
Views
2K
Solving a system of differential equations by fundamental matrix
    • Calculus and Beyond Homework Help
Replies
8
Views
1K
How to get general solution via Green's function?
    • Calculus and Beyond Homework Help
Replies
1
Views
687
Solving a DE using an ansatz
    • Calculus and Beyond Homework Help
Replies
5
Views
505
Integration via complex exponential
    • Calculus and Beyond Homework Help
Replies
2
Views
2K
Avoid unpleasant integrals in solving IVP
    • Calculus and Beyond Homework Help
Replies
3
Views
550
Diff. Eq. — Identifying Particular Solution Given solution family
    • Calculus and Beyond Homework Help
Replies
4
Views
926

Hot Threads

Recent Insights

Back
Top