Express \[\begin{aligned} x_1' &= 2tx_1 + e^tx_2\\ x_2' &= 3x_1 - 3t x_2\\\end{aligned}\] with \(x_1(0) = -5\) and \(x_2(0) = 2\) as a vector equation with a vector initial condition.
Consider the system \[\begin{aligned} x' &= 2x+y - z\\ y' &= x-3y+5z\\ z' &= 4x -7y +z.\end{aligned}\] Write this system as a vector equation.
Consider the vector equation \({\bf x}' = \begin{pmatrix} 4t & 6t^2 \\ 2 & t^3 \end{pmatrix} {\bf x} + \begin{pmatrix} e^t \\ e^{-t}\end{pmatrix}\). Write this equation as a system of 2 equations.
For Problems 4-7 recast the higher order linear differential equations as a linear system of first order equations. Find the coefficient matrix \(\ABld(t)\) and forcing \(\fBld(t)\). If the problem is an initial value problem, then be sure to state the initial condition.
\(u'' + 3tu = e^t\)
\(y''' + 2y'' -ty' + y = 0\)
\((1+t^2)y^{\prime\prime\prime} + e^{t^2}y^{\prime\prime} - \sin{(t)}y = \cos{(t)}\)
\(y^{(4)} + t^2y^{\prime\prime\prime} +\cos{(t)}y^{\prime\prime} + t^2\sin{(t)}y = te^t\)
Consider the second order equation \(y'' + p(t)y' + q(t)y = g(t)\) with initial conditions \(y'(0) = 1\) and \(y(0) = 2\). Let \(x_1 = y\) and \(x_2 = y'\), and then express this second order equation as a system of two first order equations. Be sure to include the initial condition for your system.
Consider the nth order equation \(y^{(n)} + a_1(t)y^{(n-1)} + a_2(t)y^{(n-2)}+ \dots + a_{n-1}(t)y' + a_n(t) y= g(t)\) with initial conditions \(y^{(i)}(0) = b_{i+1}\) for \(i = 0, \dots, n-1\). Express this nth order equation as a system of \(n\) first order equations. Be sure to include the initial condition for your system.
Two hanging pendula of length \(\ell\) and masses \(m_1\) and \(m_2\) are coupled by a spring. Let \(\theta_1\) and \(\theta_2\) be the angular displacement of each pendulum from its rest position. For small angles, the equations of motion are approximated by the following linear system: \[\begin{aligned} m_1\ell\theta_1^{\prime\prime} &= -m_1g\theta_1 - k\ell (\theta_1 - \theta_2) \\ m_2\ell\theta_2^{\prime\prime} &= -m_2g\theta_2 + k\ell (\theta_1 - \theta_2) \\ \end{aligned}\] Write this as a first-order linear system. Find the corresponding coefficient matrix \(\ABld\) and the forcing \(\fBld\).
For problems 11-12 determine the largest interval where a unique solution exists for the following initial value problems for first order systems.
\[{\bf x}' = \begin{pmatrix} \log(t+1) & \sqrt{9 - t^2}\\ \tan(t) & t^2\\ \end{pmatrix}{\bf x} + \begin{pmatrix} \frac{1}{t-5}\\ \sin(t) \\ \end{pmatrix} , \qquad {\bf x}(0) = \begin{pmatrix} 1\\-1\\ \end{pmatrix}\]
\[{\bf x}' = \begin{pmatrix} \log(t) & t\log(t)\\ e^{2t} & t+1\\ \end{pmatrix}{\bf x} + \begin{pmatrix} \frac{1}{t-2}\\ \sin(t) \\ \end{pmatrix} , \qquad {\bf x}(1) = \begin{pmatrix} 0\\7\\ \end{pmatrix}\]
Consider the differential system \[\frac{\dee }{\dt} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 3 & 1 \\ 2 & 2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}\]
Show that \(\begin{pmatrix}e^{4t}\\e^{4t}\end{pmatrix}\) and \(\begin{pmatrix}-e^{t}\\ 2e^{t}\end{pmatrix}\) are both solutions to this system.
Give a fundamental matrix for this system.
Give a general solution to this system in vector form.
Compute the natural fundamental matrix for this system associated with \(t=0\).
Solve the initial value problem for this system with \(x(0) = -2\) and \(y(0)= 3\).
Consider the differential system \[\frac{\dee}{\dt} \begin{pmatrix} x \\ y \end{pmatrix}= \begin{pmatrix} 1 & -\cos{(t)}\\ \cos{(t)} & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}\]
Show that both \(\begin{pmatrix}e^t\cos(\sin(t))\\ e^t\sin(\sin(t))\end{pmatrix}\) and \(\begin{pmatrix}-e^t\sin(\sin(t))\\ e^t\cos(\sin(t))\end{pmatrix}\) are both solutions to this system.
Give a fundamental matrix for this system.
Write a general solution to this system in vector form.
Compute the natural fundamental matrix of this system associated with \(t=0\).
Solve the initial value problem for this system with \(x(0) = 1\), \(y(0) = 0\).
Consider the differential system \[\frac{\dee}{\dt} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2t & -e^{t^2} \\ e^{-t^2} & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}\]
Show that both \(\begin{pmatrix}e^{t^2}\cos{(t)} \\ \sin{(t)}\end{pmatrix}\) and \(\begin{pmatrix}e^{t^2}\sin{(t)} \\ -\cos{(t)} \end{pmatrix}\) are solutions to the above system.
Give a fundamental matrix for this sytem.
Write a general solution to this system in vector form.
Compute the natural fundamental matrix of this system associated with \(t=0\).
Solve the initial value problem for this system with \(x(0) = -1\), \(y(0) = 3\).
Consider the differential system \[\frac{\dee}{\dt} \begin{pmatrix} x \\ y \end{pmatrix} =\frac{1}{1+t^2} \begin{pmatrix} (1+t)^2 & 2(1+t^2)^2\\ 2 & 1+t^2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}\]
Show that both \(\begin{pmatrix}(1+t^2)e^{3t} \\ e^{3t}\end{pmatrix}\) and \(\begin{pmatrix}(1+t^2)e^{-t} \\ -e^{-t} \end{pmatrix}\) are solutions to the above system.
Give a fundamental matrix for this sytem.
Write a general solution to this system in vector form.
Compute the natural fundamental matrix of this system associated with \(t=0\).
Solve the initial value problem for this system with \(x(0) = -4\), \(y(0) = 2\).
For problems 17-20, show that the following vector valued solutions to a linear system form a fundamental set. Find the linear system that they solve.
\(\xBld_1(t) = \begin{pmatrix}\cos(t)\\ \sin(t)+\cos(t)\end{pmatrix}, \quad \xBld_2(t) = \begin{pmatrix} -\sin(t) \\ \cos(t)-\sin(t)\end{pmatrix}\)
\(\xBld_1(t) = \begin{pmatrix}e^t\\ e^t \end{pmatrix},\quad \xBld_2(t) = \begin{pmatrix} -e^{-2t} \\ 3e^{-2t} \end{pmatrix}\)
\(\xBld_1(t) = \begin{pmatrix}2te^{2t}\\ 1-te^{2t}\end{pmatrix}, \quad \xBld_2(t) = \begin{pmatrix} -2e^{-3t} \\ e^{-3t}\end{pmatrix}\)
\(\xBld_1(t) = \begin{pmatrix}1\\ 0 \\ 2\end{pmatrix}e^{3t}, \quad \xBld_2(t) = \begin{pmatrix} t+1 \\ 1 \\ 2t+2\end{pmatrix}e^{3t}, \quad \xBld_3(t) = \begin{pmatrix}\frac{1}{2}t^2 + t + 1\\ t+1 \\ (t+1)^2\end{pmatrix}e^{3t}\).
Let \(\PhiBld(t)\) be the natural fundamental matrix associated to \(t_I\) for the system \[\frac{\dee \xBld}{\dt} = \ABld\xBld,\] where \(\ABld\) is a constant matrix. Show that \(\PhiBld(t)\) and \(\ABld\) commute, that is \[\PhiBld(t)\ABld = \ABld\PhiBld(t).\] Hint: Show that \(\PhiBld(t)\ABld\) show that \(\ABld\PhiBld(t)\) solve the same initial value problem.
Let \(\PsiBld(t)\) be a fundamental matrix for the system \[\frac{\dee \xBld}{\dt} = \ABld(t)\xBld.\] For any constant matrix \(\CBld\) such that \(\det (\CBld) \neq 0\), show that \(\PsiBld \CBld\) is also a fundamental matrix, but that \(\CBld\PsiBld\) may not be. How must \(\CBld\) and \(\ABld(t)\) be related so that \(\CBld\PsiBld\) becomes a fundamental matrix?
Let \(\PsiBld_1\) and \(\PsiBld_2\) be two fundamental matrices of \[\frac{\dee \xBld}{\dt} = \ABld(t)\xBld.\] Show that there exists a constant \(\CBld\), \(\det{(\CBld)} \neq 0\), such that \(\PsiBld_1 = \PsiBld_2\CBld\).
Hint: Show that \((\PsiBld_2^{-1}\PsiBld_1)^\prime = \ZeroBld\), and use the product rule for matrices \((\ABld\BBld)^\prime = \ABld^\prime\BBld + \ABld\BBld^\prime\) (see the exercises in the supplement on matrices and vectors).
The previous problem implies that if \(\PsiBld_1\) and \(\PsiBld_2\) are fundamental matricies of \[\frac{\dee \xBld}{\dt} = \ABld(t)\xBld,\] then their Wronskians \(W_1(t)= \det{(\PsiBld_1(t))}\), \(W_2(t) = \det{(\PsiBld_2(t))}\) differ by a constant multiple \(c\neq 0\), since \[W_1(t) = \det{(\PsiBld_1(t))} = \det{(\CBld)}\det{(\PsiBld_2(t))} = cW_2(t).\] Use Liouville’s Wronskian Theorem to reach this same conclusion.
Consider the homogeneous \(n^{\mathrm th}\) order linear differential equation in normal form \[\frac{\dee^n y}{\dt} + a_1(t)\frac{\dee^{n-1}y}{\dt}+\ldots+a_{n-1}(t)\frac{\dee y}{\dt} + a_n(t)y = 0,\] and its equivalent \(n\)-dimensional first-order linear system \[\frac{\dee \xBld}{\dt} = \ABld(t)\xBld.\] Show Abel’s Wronskian Theorem for the \(n^{\mathrm th}\) order equation using Liouville’s Wronskian Theorem for the first order linear system.