All Questions
13
questions
0
votes
1
answer
24
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Finding suitable $x:[-T,T]\to\mathbb{R}^n, A\in\mathbb{R}^{n\times n}$ such that $x'(t) = Ax(t)$ when $\frac{d^2}{dt^2}B(t)=B(t)$
Suppose that $\frac{d^2}{dt^2}B(t) = B(t)$ for some matrix $B$ when $t\in [-T, T], T > 0$. I am tasked to determine suitable $x:[-T,T]\to\mathbb{R}^n, A\in\mathbb{R}^{n\times n}$ such that $x'(t) = ...
- 1,221
5
votes
2
answers
429
views
Exponential growth of a cow farm with constraints in Minecraft
This question is distinct from Exponential growth of cow populations in Minecraft in that an important constraint present in Minecraft is missing from that post. Here are the following constraints:
...
- 861
0
votes
2
answers
1k
views
How to find an equation describing mass of sample over time, radioactive decay?
If you have an initial mass of 50kg for a radioactive sample, which has a half-life of 5000 years, how would you go about finding an equation that describes the mass of the sample over time??
My ...
- 141
0
votes
6
answers
132
views
Showing Linear Independence of $\, \{e^x, \, e^{-x}\} $
We examine the equation
$$ae^x + be^{-x} = 0$$
My book say manipulate to get
$$ae^{2x} = -b$$
and "The two members are identical for all $x$ only if
$$a =b = 0 \, \text{ "}$$
[Therefore the set ...
- 4,320
0
votes
2
answers
333
views
The solution of $dX(t)/dt = A X(t) + X(t) A + Q$?
According to the perfect answer from A.Γ. in Any compact solution for $dX/dt = A X(t) + X(t) A^T$?, I understand that the solution of
$$\frac{dX(t)}{dt} = AX(t) + X(t)A^T,$$
where $A, X(t) \in {\...
- 665
0
votes
1
answer
247
views
How to compute the exponential of this matrix?
I am trying to prove all the results regarding linear algebra in my ODE class. I have already convinced myself that if I have a matrix $T$ which has an eigenvalue $\lambda = a + ib$ and an associated ...
1
vote
1
answer
41
views
Exponential of a non terminating matric
So I understand how to calculate the exponential of matrices that eventually terminate; however, how to approach the cases in which the matrix does not seem to truncate? For example with the matrix $M=...
- 51
4
votes
5
answers
606
views
Solve: $x''(t)-2x'(t) + x(t) = 2 \sin(3t)$
It is asked to solve the ODE $x''(t)-2x'(t) + x(t) = 2 \sin(3t)$ for $x(0)=10, \; x'(0)=0$
It is equivalent to the first order system in two variables
$$\begin{bmatrix} x' \\ y' \end{bmatrix} = \...
- 3,207
0
votes
1
answer
56
views
Solve explicitly for time from a sum of exponentials
Suppose $f(t) = 0$. How can I solve for time, $t$, in the following expression.
$f(t) = k_1{e}^{- \alpha t} + k_2{e}^{- \beta t} + k_3{e}^{- \gamma t}\left( k_4 sin(\omega_d t) + k_5 cos (\omega_d t) ...
- 1
0
votes
3
answers
121
views
Exponentiation of a $2\times 2$ matrix
We know:
$$\exp(At)=I+ \sum^{\infty}_{n=1}\frac{A^nt^n}{n!}$$
Here $$A= \begin{pmatrix} 0 & 1 \\ -w^2 & 0\end{pmatrix}$$ is a $2\times 2$ matrix,
$I$ is identity matrix.
How to show:
$$\...
- 1,919
2
votes
3
answers
123
views
If $A$ is a $2\times 2$ matrix with a repeated eigenvalue $r$, then $\mathrm{e}^{At}=\mathrm{e}^{rt}\left[I+(A-rI)t\right]$
If $A$ is a $2\times 2$ matrix with a repeated eigenvalue $r$, show that $\mathrm{e}^{At}=\mathrm{e}^{rt}\left[I+(A-rI)t\right]$.
I have already been able to show that if $A$ is an arbitrary $2\...
- 1,369
0
votes
1
answer
1k
views
Problem with commutator relations
part a) is fine. part b) is not.
A commutator is defined as, for operators $A$ and $B$, $[A,B]=AB-BA$.
- 5,244
2
votes
1
answer
4k
views
Fundamental matrix and exponential of matrix using Laplace Transform
I'm trying to work out how to find $$\exp(At)$$ for a system of linear differential equations $$x'=Ax.$$
I know that the solution is a fundamental matrix of the system such that $$\exp(At)=I$$
at ...
- 1,146