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
asked 2021-02-21
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.
\(\displaystyle{\left[\lambda_{{{1}}}=-{1}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}{0}{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},\lambda_{{{2}}}={3}{i}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}-{i}{1}+{i}{7}{i}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},\lambda_{{3}}=-{3}{i}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}+{i}{1}-{i}-{7}{i}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace}\right]}\)

Answers (1)

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:
\(\displaystyle{y}_{{1}}={c}_{{1}}{e}^{{{t}}}{\left({c}_{{2}}{\left({2}{\sin{{\left({3}{t}\right)}}}-{\cos{{\left({3}{t}\right)}}}\right)}+{c}_{{3}}{\left({2}{\cos{{\left(-{3}{t}\right)}}}-{\sin{{\left(-{3}{t}\right)}}}\right)}\right)}\)
\(\displaystyle{y}_{{2}}={c}_{{2}}{\left({\sin{{\left({3}{t}\right)}}}+{\cos{{\left({3}{t}\right)}}}\right)}+{c}_{{3}}{\left({\cos{{\left(-{3}{t}\right)}}}+{\sin{{\left(-{3}{t}\right)}}}\right)}{)}\)
\(\displaystyle{y}_{{3}}={3}{c}_{{1}}{e}^{{{t}}}{c}_{{2}}{\left({2}{\sin{{\left({3}{t}\right)}}}+{\cos{{\left({3}{t}\right)}}}\right)}+{7}{c}_{{3}}{\sin{{\left(-{3}{t}\right)}}}{)}\)
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