# Tagged Questions

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### Derivative with respect to a function

We have a function ${f(s,{\psi(s)}_{3\times 1})}_{3\times1}\tag1$ Given Data $f,\psi$ are matrices and their dimensions are already given in the question s is not a matrix, it is a scalar ...
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### Find $e^{AT}$ where $A$ is a Matrix that is given

How to find the value of $e^{At}$ where $A$ is the matrix $A =\begin{bmatrix} 4 & 3 \\ 2 & -1 \end{bmatrix}$
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### Drawing phase portrait

This is the question in my textbook. I am a bit lost for 3 hours now. Could anyone please point me to the right direction? Let the $2 \times 2$ matrix $A$ have real, distinct eigenvalues $\lambda$ ...
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### Deducing the exact solution of a ODE

In page 53 of Arieh Iserles's A first course in the numerical analysis of differential equations, he presents the following ODE: $(\vec{y})'=\Gamma\cdot\vec{y}$, $\vec{y}(0)=\vec{y_0}$ Using the ...
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### How to compute time ordered Exponential?

So say you have a matrix dependent on a variable t: $$A(t)$$ how do you compute $$e^{A(t)}$$ It seems Sylvester's formula, my standard method of computing matrix exponentials can't be applied ...
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### Show that $e^{t(A+B)} = e^{tA}e^{tB}$ for all $t \in \mathbb{R}$ if, and only if $AB = BA$.

Let A,B real or complex matrixes. Show that $e^{t(A+B)} = e^{tA}e^{tB}$ for all $t \in \mathbb{R}$ if, and only if $AB = BA$. I demonstrated the reciprocal: $\Leftarrow )$ The two equations are ...
Let $A$ be a complex $n \times n$ matrix, such that the function $t\mapsto e^{tA}x$ is bounded on $\mathbb{R}$ and nonzero, for some vector $x\in \mathbb{C}$. How can we prove that $\inf_{t\in ... 0answers 28 views ### 3-Species Population Model I am trying to solve a 3-species predator-prey system in matlab. Here is the equation: $$\frac{d}{dt} \begin{bmatrix} N_1 \\ N_2 \\ N_3 \\ \end{bmatrix} = \begin{bmatrix} N_1 & 0 & 0 \\ 0 ... 1answer 54 views ### Differential equation of the form y'=Ay+b(x) with b(x)=(\sin{(\omega x)},0) I have a question regarding the following specific differential equation.$$y'=\left(\begin{matrix} 0 & 1 \\ -1 & 0 \\ \end{matrix}\right)y+\left(\begin{matrix} ... 0answers 30 views ### Find det M in the differential equation Let's take a scalar equation:$\left(\tan{t}\right)y'''+2y''-\left(\sin{t}\right)y'+2y=0$Defined on the interval$ t \in (-\frac{\pi}{2}, \frac{\pi}{2})$Let$M(t,\frac{\pi}{4})$be a solving matrix ... 1answer 36 views ### Proof of Lyapunov Stability for Constant Matrix System I am trying to find the necessary and sufficient conditions for the point of equilibrium x=0 of$x'=Ax$to be Lyapunov stable, where A is constant matrix. The book I'm using briefly touches on this, ... 2answers 91 views ### Show each eigenvalue of a companion matrix has geometric multiplicity$=1$. Given the differential equation $$x^{(n)}(t)+c_{n-1}x^{(n-1)}(t) + \dotsb + c_1x'(t) + c_0=0,$$ we can form a vector$\xi = (x, x', \dotsc, x^{(n-1)})$, and then we have $$\xi'(t) = A\xi,$$ where$A$... 0answers 31 views ### Matrix differential equation MX' = AX+XB+C(t) Here is matrix differential equation: $$\mu \frac {dX}{dt}=AX+XB+C(t)$$ $$X(0) = X_0$$ Here$\mu$is real diagonal matrix,$X$is$m$by$n$matrix.$A$,$B$are real square matrices of constant ... 0answers 36 views ### What are the different solution concepts for Matrix-Ordinary Differential Equation [Theory Question] I was recently given a ODE to solve from a boss at work, with the knowledge that I haven't done them before and this will help me learn. I've spent 10 hours so far learning the basics of ODEs. The ... 2answers 84 views ### Minimizing the following objective function with matrices Suppose$A$and$B$are known matrices, and we are to find matrix$X$that minimizes the following function, $$\frac{1}{2}||X||^2+\frac{1}{2}||X^TAX-B||^2$$ Taking the relevant derivative w.r.t$X$... 0answers 44 views ### Finding Fundamental Matrix of Differential Equations with Periodic Coefficients So I am supposed to find the fundamental matrix of$x'_1=(1+2\cos(2t))x_1+(1-2\sin(2t))x_2x'_2=-(1+2\sin(2t)x_1+(1-2\cos(2t))x_2$The book suggests using the transformation$x=Q(t)\hat{x}$... 3answers 32 views ### Determinant of solution matrix Let$\phi(t)$be a solution matrix. Show that $$\det\phi(t)=\det\phi(t)\exp\int_{t_0}^t\sum_{j=1}^na_{jj}(s)\,ds.$$ I know that$[\det\phi(t)]'=\sum_{j=1}^na_{jj}(t)\det\phi(t),$but I am not how to ... 1answer 49 views ### Second Order Derivative of a function$f:R^2\to R^2$The Exercise: My Work: Part 1: $$Df=\left( \begin{array}{ccc} D_1f_1 & D_2f_1\\ D_1f_2 & D_2f_2 \\\end{array} \right)$$ $$f_1(x,y)=\sin x+\sin y$$ $$f_2(x,y)=\cos x+\cos y$$ $$... 1answer 81 views ### Find the determinant of a solving matrix I have such ODE:$$\frac{dy}{dt}=\begin{pmatrix} \sin^2t & e^{-t} \\ e^t & \cos^2t \end{pmatrix} y=A(t)y(t)$$and let M(t,1) be the solving matrix (a matrix whose columns ... 2answers 79 views ### Why can't you swap rows in the matrix for a system of linear differential equations? If you are given a Matrix A, and then asked to solve the initial value problem x'=Ax, why can one not swap rows before starting the problem. I tried it with a 3x3 matrix on wolfram alpha and got two ... 1answer 26 views ### Fundamental Matrix with Sums Let$$\Phi(t)=\begin{bmatrix} x_{11}(t) & x_{12}(t)\\ x_{21}(t) & x_{22}(t) \end{bmatrix} $$be a fundamental matrix for$$x'=A(t)$$where$$A=\begin{bmatrix} a_{11}(t) & a_{12}(t)\\ ... 1answer 49 views ### Integrating a Matrix in differential-equations Find $$\int_0^t A(s)ds$$ if $$A(t)=\begin{pmatrix}\sin(t),\cos(t)\\ -\sin(t),\cos(t)\end{pmatrix}$$ I'm a little confused with the format of the question because it asks me to integrate with respect ... 1answer 43 views ### number of solutions of a system of linear equations Consider a system$\Sigma (y)$of$m$linear equations in$n$variables$x_1,\cdots,x_n$:$\sum_{j=1}^n a_{i,j}(y)\cdot x_j=b_i(y)$,$i=1,\cdots,m$, whose coefficients$a_{i,j}(y)$and$b_i(y)$are ... 2answers 73 views ### how to compute the determinant of a linear map Let$V$be the vector space of$m\times n$matrices over a field$F$. Fix an$m\times m$matrix$A$and an$n\times n$matrix$C$, and consider the map$\phi: V\longrightarrow V\$ defined by ...
Question is to find the fundamental matrix(F(t)) satisfying F(0)=I for the given system of equation below. $$x' =\left(\begin{array}{rr}2 & 3 \\ -1 & -2\end{array}\right)x$$ My solution ...