# The coefficient matrix for a system of linear differential equations of the form y^{1} = A_{y} has the given eigenvalues and eigenspace bases. Find the general solution for the system. \left[\lambda_{1}=-1\Rightarrow\left\{\begin{bmatrix}1 0 3 \end{bmatrix}\right\},\lambda_{2}=3i\Rightarrow\left\{\begin{bmatrix}2-i 1+i 7i \end{bmatrix}\right\},\lambda_3=-3i\Rightarrow\left\{\begin{bmatrix}2+i 1-i -7i \end{bmatrix}\right\}\right]

Question
Differential equations
The coefficient matrix for a system of linear differential equations of the form $$\displaystyle{y}^{{{1}}}={A}_{{{y}}}$$ has the given eigenvalues and eigenspace bases. Find the general solution for the system.
$$\displaystyle{\left[\lambda_{{{1}}}=-{1}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}{0}{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},\lambda_{{{2}}}={3}{i}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}-{i}{1}+{i}{7}{i}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},\lambda_{{3}}=-{3}{i}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}+{i}{1}-{i}-{7}{i}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace}\right]}$$

2021-02-22
Step 1
The general solution is
$$\displaystyle{y}={c}_{{{1}}}{y}_{{{1}}}+{c}_{{{2}}}{y}_{{{2}}}+{c}_{{{3}}}{y}_{{{3}}}$$
for $$\displaystyle{y}_{{1}}={e}^{{\lambda_{{1}}{t}}}{u}$$
$$\displaystyle{y}_{{2}}={e}^{{{a}{t}}}{\left({\sin{{\left({b}{t}\right)}}}{R}{e}{\left({u}\right)}+{\cos{{\left({b}{t}\right)}}}{I}{m}{\left({u}\right)}\right)}$$
$$\displaystyle{y}_{{3}}={e}^{{{a}{t}}}{\left({\cos{{\left({b}{t}\right)}}}{R}{e}{\left({u}\right)}-{\sin{{\left({b}{t}\right)}}}{I}{m}{\left({u}\right)}\right)}$$
Step 2
We have
$$\displaystyle{\left[\lambda_{{1}}=-{1}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}{0}{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},\lambda_{{2}}={3}{i}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}-{i}{1}+{i}{7}{i}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},\lambda_{{3}}=-{3}{i}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}+{i}{1}-{i}-{7}{i}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace}\right]}$$
Step 3
Then
$$\displaystyle{y}_{{1}}={e}^{{-{t}}}{\left({b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{103}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\right)}$$
$$\displaystyle{y}_{{2}}={e}^{{{0}{t}}}{b}{i}{g}{g{{\left({\left({\sin{{\left({3}{t}\right)}}}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{212}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}+{\cos{{\left({3}{t}\right)}}}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}-{1}{11}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{b}{i}{g}{g}\right)}\right.}}}$$
$$\displaystyle={b}{i}{g}{g{{\left({\left({\sin{{\left({3}{t}\right)}}}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{212}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}+{\left({\cos{{\left({3}{t}\right)}}}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}-{111}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{b}{i}{g}{g}\right)}\right.}\right.}}}$$
$$\displaystyle{y}_{{3}}={e}^{{{0}{t}}}{b}{i}{g}{g{{\left({\cos{{\left(-{3}{t}\right)}}}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{210}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}-{\sin{{\left(-{3}{t}\right)}}}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}-{17}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{b}{i}{g}{g}\right)}}}$$
$$\displaystyle={b}{i}{g}{g{{\left({\cos{{\left(-{3}{t}\right)}}}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{210}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}-{\sin{{\left(-{3}{t}\right)}}}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}-{17}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{b}{i}{g}{g}\right)}}}$$
Step 4
Hence the general solution is
$$\displaystyle{y}={c}_{{1}}{y}_{{1}}+{c}_{{2}}{y}_{{2}}+{c}_{{3}}{y}_{{3}}$$
$$\displaystyle={c}_{{1}}{e}^{{-{t}}}{\left({b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{103}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\right)}+{c}_{{2}}{b}{i}{g}{g{{\left({\left({\sin{{\left({3}{t}\right)}}}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{212}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}+{\left({\cos{{\left({3}{t}\right)}}}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}-{111}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{b}{i}{g}{g}\right)}+{c}_{{3}}{b}{i}{g}{g{{\left({\cos{{\left(-{3}{t}\right)}}}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{210}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}-{\sin{{\left(-{3}{t}\right)}}}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}-{17}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{b}{i}{g}{g}\right)}}}\right.}\right.}}}$$
Step\ 5
The individual functions are
$$\displaystyle{y}_{{1}}={c}_{{1}}{e}^{{{t}}}{\left({c}_{{2}}{\left({2}{\sin{{\left({3}{t}\right)}}}-{\cos{{\left({3}{t}\right)}}}\right)}+{c}_{{3}}{\left({2}{\cos{{\left(-{3}{t}\right)}}}-{\sin{{\left(-{3}{t}\right)}}}\right)}\right)}$$
$$\displaystyle{y}_{{2}}={c}_{{2}}{\left({\sin{{\left({3}{t}\right)}}}+{\cos{{\left({3}{t}\right)}}}\right)}+{c}_{{3}}{\left({\cos{{\left(-{3}{t}\right)}}}+{\sin{{\left(-{3}{t}\right)}}}\right)}{)}$$
$$\displaystyle{y}_{{3}}={3}{c}_{{1}}{e}^{{{t}}}{c}_{{2}}{\left({2}{\sin{{\left({3}{t}\right)}}}+{\cos{{\left({3}{t}\right)}}}\right)}+{7}{c}_{{3}}{\sin{{\left(-{3}{t}\right)}}}{)}$$
Result:
$$\displaystyle{y}_{{1}}={c}_{{1}}{e}^{{{t}}}{\left({c}_{{2}}{\left({2}{\sin{{\left({3}{t}\right)}}}-{\cos{{\left({3}{t}\right)}}}\right)}+{c}_{{3}}{\left({2}{\cos{{\left(-{3}{t}\right)}}}-{\sin{{\left(-{3}{t}\right)}}}\right)}\right)}$$
$$\displaystyle{y}_{{2}}={c}_{{2}}{\left({\sin{{\left({3}{t}\right)}}}+{\cos{{\left({3}{t}\right)}}}\right)}+{c}_{{3}}{\left({\cos{{\left(-{3}{t}\right)}}}+{\sin{{\left(-{3}{t}\right)}}}\right)}{)}$$
$$\displaystyle{y}_{{3}}={3}{c}_{{1}}{e}^{{{t}}}{c}_{{2}}{\left({2}{\sin{{\left({3}{t}\right)}}}+{\cos{{\left({3}{t}\right)}}}\right)}+{7}{c}_{{3}}{\sin{{\left(-{3}{t}\right)}}}{)}$$

### Relevant Questions

The coefficient matrix for a system of linear differential equations of the form $$y^1=Ay$$
has the given eigenvalues and eigenspace bases. Find the general solution for the system
$$\lambda_1=2i \Rightarrow \left\{ \begin{bmatrix}1+i\\ 2-i \end{bmatrix} \right\} , \lambda_2=-2i \Rightarrow \left\{ \begin{bmatrix}1-i\\ 2+i \end{bmatrix} \right\}$$
The coefficient matrix for a system of linear differential equations of the form $$y^1=Ay$$
=Ay has the given eigenvalues and eigenspace bases. Find the general solution for the system \lambda_1=3 \Rightarrow \left\{ \begin{bmatrix}1\\1\\0 \end{bmatrix} \right\} , \lambda_2=0 \Rightarrow \left\{ \begin{bmatrix}1\\5\\1 \end{bmatrix} , \begin{bmatrix}2\\1\\4 \end{bmatrix} \right\}
The coefficient matrix for a system of linear differential equations of the form $$y_1=Ay$$
has the given eigenvalues and eigenspace bases. Find the general solution for the system
$$\lambda_1=3+i \Rightarrow \left\{ \begin{bmatrix}2i\\ i \end{bmatrix} \right\} , \lambda_2=3-i \Rightarrow \left\{ \begin{bmatrix}-2i\\ -i \end{bmatrix} \right\}$$
The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution.
$$\displaystyle{\left[\begin{matrix}{1}&{0}&-{1}&-{2}\\{0}&{1}&{2}&{3}\end{matrix}\right]}$$
The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution.
$$\displaystyle{\left[\begin{matrix}{1}&-{1}&{0}&-{2}&{0}&{0}\\{0}&{0}&{1}&{2}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}\end{matrix}\right]}$$
The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution. Write the solution in vector form. $$\displaystyle{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&{0}&−{1}&{3}&{9}\backslash{0}&{1}&{2}&−{5}&{8}\backslash{0}&{0}&{0}&{0}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}$$
Consider the bases $$\displaystyle{B}={\left({b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}\backslash{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{3}\backslash{5}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\right)}{o}{f}{R}^{{2}}{\quad\text{and}\quad}{C}={\left({b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{1}\backslash{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{0}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}\backslash{1}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\right)}{o}{f}{R}^{{3}}$$.
and the linear maps PSKS \in L (R^2, R^3) and T \in L(R^3, R^2) given given (with respect to the standard bases) by $$\displaystyle{\left[{S}\right]}_{{{E},{E}}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}&-{1}\backslash{5}&-{3}\backslash-{3}&{2}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{\quad\text{and}\quad}{\left[{T}\right]}_{{{E},{E}}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&-{1}&{1}\backslash{1}&{1}&-{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}$$ Find each of the following coordinate representations. $$\displaystyle{\left({a}\right)}{\left[{S}\right]}_{{{B},{E}}}$$
The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution. Write the solution in vector form. $$\begin{bmatrix}1&-2&0&0&-3\\0&0&1&0&-4\\0&0&0&1&5\end{bmatrix}$$
The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution. Write the solution in vector form. $$\begin{bmatrix}1&3&0&-2&6\\0&0&1&4&7\\0&0&0&0&0\end{bmatrix}$$
$$\displaystyle{\left\lbrace\begin{matrix}{x}+{y}={0}\\{5}{x}-{2}{y}-{2}{z}={12}\\{2}{x}+{4}{y}+{z}={5}\end{matrix}\right.}$$