121
Term Test 2 / You may post solutions
« on: November 19, 2019, 04:25:47 AM »
After an (almost) perfect solution is posted, no need to post the same solution
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122
Term Test 2 / Problem 4 (noon)
« on: November 19, 2019, 04:24:35 AM »
Find the general real solution to
$$
\mathbf{x}'=\begin{pmatrix}
1 & 3\\
-2 &-3\end{pmatrix}\mathbf{x}
$$
and sketch trajectories.
$$
\mathbf{x}'=\begin{pmatrix}
1 & 3\\
-2 &-3\end{pmatrix}\mathbf{x}
$$
and sketch trajectories.
123
Term Test 2 / Problem 4 (morning)
« on: November 19, 2019, 04:23:46 AM »
Find the general real solution to
$$
\mathbf{x}'=\begin{pmatrix}
2 & -3\\
4 &-2\end{pmatrix}\mathbf{x}
$$
and sketch trajectories.
$$
\mathbf{x}'=\begin{pmatrix}
2 & -3\\
4 &-2\end{pmatrix}\mathbf{x}
$$
and sketch trajectories.
124
Term Test 2 / Problem 4 (main sitting)
« on: November 19, 2019, 04:23:20 AM »
Find the general real solution to
$$
\mathbf{x}'=\begin{pmatrix}
3 & 3\\
-2 &-1\end{pmatrix}\mathbf{x}
$$
and sketch trajectories.
$$
\mathbf{x}'=\begin{pmatrix}
3 & 3\\
-2 &-1\end{pmatrix}\mathbf{x}
$$
and sketch trajectories.
125
Term Test 2 / Problem 3 (noon)
« on: November 19, 2019, 04:22:43 AM »
(a) Find the general solution of
$$
\mathbf{x}'=\begin{pmatrix} 1 &2\\
1 &0\end{pmatrix}\mathbf{x}
$$
and sketch trajectories.
(b) Find the general solution
$$
\mathbf{x}'=\begin{pmatrix} 1 &2\\
1 &0\end{pmatrix}\mathbf{x}+
\begin{pmatrix} 0 \\[1pt]
\dfrac{6 e^{3t }}{e^{2t}+1}\end{pmatrix}.$$
$$
\mathbf{x}'=\begin{pmatrix} 1 &2\\
1 &0\end{pmatrix}\mathbf{x}
$$
and sketch trajectories.
(b) Find the general solution
$$
\mathbf{x}'=\begin{pmatrix} 1 &2\\
1 &0\end{pmatrix}\mathbf{x}+
\begin{pmatrix} 0 \\[1pt]
\dfrac{6 e^{3t }}{e^{2t}+1}\end{pmatrix}.$$
126
Term Test 2 / Problem 3 (morning)
« on: November 19, 2019, 04:21:57 AM »
(a) Find the general solution of
$$
\mathbf{x}'=\begin{pmatrix} -2 &1\\
-1 &0\end{pmatrix}\mathbf{x}
$$
and sketch trajectories.
(b) Find the general solution
$$
\mathbf{x}'=\begin{pmatrix} -2 &1\\
-1 &0\end{pmatrix}\mathbf{x}+
\begin{pmatrix} 0 \\
\dfrac{e^{-t}} {t^2+1} \end{pmatrix}.
$$
$$
\mathbf{x}'=\begin{pmatrix} -2 &1\\
-1 &0\end{pmatrix}\mathbf{x}
$$
and sketch trajectories.
(b) Find the general solution
$$
\mathbf{x}'=\begin{pmatrix} -2 &1\\
-1 &0\end{pmatrix}\mathbf{x}+
\begin{pmatrix} 0 \\
\dfrac{e^{-t}} {t^2+1} \end{pmatrix}.
$$
127
Term Test 2 / Problem 3 (main sitting)
« on: November 19, 2019, 04:21:01 AM »
(a) Find the general solution of
$$
\mathbf{x}'=\begin{pmatrix} 1 &1\\
-2 &4\end{pmatrix}\mathbf{x}
$$
classify fixed point $(0,0)$ and sketch trajectories.
(b) Find the general solution
$$
\mathbf{x}'=\begin{pmatrix} 1 &1\\
-2 &4\end{pmatrix}\mathbf{x}+
\begin{pmatrix} \dfrac{e^{4t }}{e^{2t}+1} \\
0\end{pmatrix}.
$$
$$
\mathbf{x}'=\begin{pmatrix} 1 &1\\
-2 &4\end{pmatrix}\mathbf{x}
$$
classify fixed point $(0,0)$ and sketch trajectories.
(b) Find the general solution
$$
\mathbf{x}'=\begin{pmatrix} 1 &1\\
-2 &4\end{pmatrix}\mathbf{x}+
\begin{pmatrix} \dfrac{e^{4t }}{e^{2t}+1} \\
0\end{pmatrix}.
$$
128
Term Test 2 / Problem 2 (noon)
« on: November 19, 2019, 04:19:53 AM »
Consider equation
\begin{equation}
y'''+y''+4y'+4y=-24e^{-2t}.
\label{2-1}
\end{equation}
(a) Write a differential equation for Wronskian of $y_1,y_2,y_3$, which are solutions for homogeneous equation and solve it.
(b) Find fundamental system $\{y_1,y_2,y_3\}$ of solutions for homogeneous equation, and find their Wronskian. Compare with (a).
(c) Find the general solution of (\ref{2-1}).
\begin{equation}
y'''+y''+4y'+4y=-24e^{-2t}.
\label{2-1}
\end{equation}
(a) Write a differential equation for Wronskian of $y_1,y_2,y_3$, which are solutions for homogeneous equation and solve it.
(b) Find fundamental system $\{y_1,y_2,y_3\}$ of solutions for homogeneous equation, and find their Wronskian. Compare with (a).
(c) Find the general solution of (\ref{2-1}).
129
Term Test 2 / Problem 2 (morning)
« on: November 19, 2019, 04:18:46 AM »
Consider equation
\begin{equation}
y'''-2y''+4y'-8y=15\cos (t).
\label{2-1}
\end{equation}
(a) Write a differential equation for Wronskian of $y_1,y_2,y_3$, which are solutions for homogeneous equation and solve it.
(b) Find fundamental system $\{y_1,y_2,y_3\}$ of solutions for homogeneous equation, and find their Wronskian. Compare with (a).
(c) Find the general solution of (\ref{2-1}).
\begin{equation}
y'''-2y''+4y'-8y=15\cos (t).
\label{2-1}
\end{equation}
(a) Write a differential equation for Wronskian of $y_1,y_2,y_3$, which are solutions for homogeneous equation and solve it.
(b) Find fundamental system $\{y_1,y_2,y_3\}$ of solutions for homogeneous equation, and find their Wronskian. Compare with (a).
(c) Find the general solution of (\ref{2-1}).
130
Term Test 2 / Problem 2 (main sitting)
« on: November 19, 2019, 04:17:26 AM »
Consider equation
\begin{equation}
y'''+4y''+y'-6y=24e^{t}.
\label{2-1}
\end{equation}
(a) Write a differential equation for Wronskian of $y_1,y_2,y_3$, which are solutions for homogeneous equation and solve it.
(b) Find fundamental system $\{y_1,y_2,y_3\}$ of solutions for homogeneous equation, and find their Wronskian. Compare with (a).
(c) Find the general solution of (\ref{2-1}).
\begin{equation}
y'''+4y''+y'-6y=24e^{t}.
\label{2-1}
\end{equation}
(a) Write a differential equation for Wronskian of $y_1,y_2,y_3$, which are solutions for homogeneous equation and solve it.
(b) Find fundamental system $\{y_1,y_2,y_3\}$ of solutions for homogeneous equation, and find their Wronskian. Compare with (a).
(c) Find the general solution of (\ref{2-1}).
131
Term Test 2 / Problem 1 (noon)
« on: November 19, 2019, 04:15:32 AM »
(a) Find the general solution of
$$
y''-3y'+2y=\frac{e^{3t}}{e^{2t}+1}.
$$
(b) Find solution satisfying
$$y(0)=y'(0)=0.$$
$$
y''-3y'+2y=\frac{e^{3t}}{e^{2t}+1}.
$$
(b) Find solution satisfying
$$y(0)=y'(0)=0.$$
132
Term Test 2 / Problem 1 (morning)
« on: November 19, 2019, 04:14:47 AM »
(a) Find the general solution of
$$
y''-y=\frac{12}{e^{t}+1}.
$$
(b) Find solution satisfying
$$y(0)=y'(0)=0.$$
$$
y''-y=\frac{12}{e^{t}+1}.
$$
(b) Find solution satisfying
$$y(0)=y'(0)=0.$$
133
Term Test 2 / Problem 1 (main sitting)
« on: November 19, 2019, 04:13:51 AM »
(a) Find the general solution of
$$
y''+4y=\frac{1}{\cos^2(t)},\qquad -\frac{\pi}{2}<t<\frac{\pi}{2}.
$$
(b) Find solution satisfying
$$y(0)=y'(0)=0.$$
$$
y''+4y=\frac{1}{\cos^2(t)},\qquad -\frac{\pi}{2}<t<\frac{\pi}{2}.
$$
(b) Find solution satisfying
$$y(0)=y'(0)=0.$$
135
Chapter 7 / Re: Finding linear independence
« on: November 11, 2019, 06:58:04 AM »
Those are not just vectors, but vector-valued functions and you need to check that for constant coefficients their linear combination is identically $0$ if and only if these coefficients are $0$.
Try first to look at components of this vector-function.
Try first to look at components of this vector-function.