Science:Math Exam Resources/Courses/MATH152/April 2017/Question B 04 (c)

MATH152 April 2017

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Question B 04 (c)
A model of love affairs is described by the following system of differential equations ${\begin{aligned}R'&=R+4J,\\J'&=-2R-3J,\end{aligned}}$ where $R(t)$ and $J(t)$ are, respectively, Romeo’s and Juliet’s amount of love for each other at time $t$. (c) Express the general solution to the system in either real or complex form.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
The general solution to the system of linear first order differential equations of the form ${\frac {d{\mathbf {x}}}{dt}}=A{\mathbf {x}}$ can be written as: ${\mathbf {x}}=c_{1}{\mathbf {v}}_{1}e^{\lambda _{1}t}+c_{2}{\mathbf {v}}_{2}e^{\lambda _{2}t}$ for eigenvectors ${\mathbf {v}}_{i}$ and eigenvalues $\lambda _{i}$ of the matrix $A$, with arbitrary constants $c_{i}$.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Since the eigenvectors are linearly independent, we know that the general solution can be written as: ${\mathbf {x}}=c_{1}{\mathbf {v}}_{1}e^{\lambda _{1}t}+c_{2}{\mathbf {v}}_{2}e^{\lambda _{2}t}$ for eigenvectors ${\mathbf {v}}_{i}$ and eigenvalues $\lambda _{i}$ of the matrix $A$, with arbitrary constants $c_{i}$. From part b), we know the eigenvalues and eigenvectors for $A$. So substituting in, we get that the general solution is: Answer: $\color {blue}c_{1}{\begin{bmatrix}1\\{\frac {-1+i}{2}}\\\end{bmatrix}}e^{(-1+2i)t}+c_{2}{\begin{bmatrix}1\\{\frac {-1-i}{2}}\\\end{bmatrix}}e^{(-1-2i)t},{\text{ with arbitrary constants }}c_{1},c_{2}.$