Saltar la navegación

Evaluación

Actividad de habilidades y conocimientos 3. Resuelva los sistemas de ecuaciones diferenciales indicados en negritas.

Pregunta

\pmb{\dot{x}= \left( \begin{array}{rr} 12&-9\\ 4&0 \end{array} \right) x}

Respuestas

x=k_1\left(\begin{array}{r} 3\\ 2 \end{array}\right)e^{6t}+k_2\left(\begin{array}{r} 3\\ 2 \end{array}\right)te^{6t}

x=k_1\left(\begin{array}{r} 3\\ 2 \end{array}\right)e^{6t}+k_2\left(\begin{array}{r} 3t+2\\ 2t+1 \end{array}\right)e^{6t}

x=k_1\left(\begin{array}{r} 3\\ 2 \end{array}\right)e^{6t}+k_2\left(\begin{array}{r} 2\\ 1 \end{array}\right)te^{6t}

x=k_1\left(\begin{array}{r} 3\\ 2 \end{array}\right)e^{6t}+k_2\left(\begin{array}{r} 2\\ 1 \end{array}\right)e^{6t}

Retroalimentación

Pregunta

\pmb{\dot{x}= \left( \begin{array}{rr} -1&-2\\ 3&4 \end{array} \right) x}

Respuestas

x=k_1\left(\begin{array}{r} -2\\ 3 \end{array}\right)e^{2t}+k_2\left(\begin{array}{r} -1\\ 1 \end{array}\right)e^{t}

x=k_1\left(\begin{array}{r} -2\\ 3 \end{array}\right)e^{-2t}+k_2\left(\begin{array}{r} -1\\ 1 \end{array}\right)e^{-t}

x=k_1\left(\begin{array}{r} 2\\ -3 \end{array}\right)e^{2t}+k_2\left(\begin{array}{r} 1\\ -1 \end{array}\right)e^{t}

x=k_1\left(\begin{array}{r} -2\\ 3 \end{array}\right)e^{t}+k_2\left(\begin{array}{r} -1\\ 1 \end{array}\right)e^{2t}

Retroalimentación

Pregunta

\pmb{\dot{x}= \left( \begin{array}{rr} 1&1\\ -2&-1 \end{array} \right) x}

Respuestas

x=w_1+w_2 \quad \text{donde}\quad w_1=[k_1\cos(t)+k_2\text{sen}(t)]\left(\begin{array}{r} 1\\ -1 \end{array}\right) \quad \text{y} \quad w_2=[k_2\cos(t)+k_1\text{sen}(t)]\left(\begin{array}{r} 0\\ 1 \end{array}\right)

x=w_1+w_2 \quad \text{donde}\quad w_1=[k_1\cos(t)+k_2\text{sen}(t)]\left(\begin{array}{r} 1\\ -1 \end{array}\right) \quad \text{y} \quad w_2=[k_1\cos(t)-k_2\text{sen}(t)]\left(\begin{array}{r} 1\\ -1 \end{array}\right)

x=w_1+w_2 \quad \text{donde}\quad w_1=[k_1\cos(t)+k_2\text{sen}(t)]\left(\begin{array}{r} 1\\ -1 \end{array}\right) \quad \text{y} \quad w_2=[k_2\text{sen}(t)+k_1\cos(t)]\left(\begin{array}{r} 1\\ -1 \end{array}\right)

x=w_1+w_2 \quad \text{donde}\quad w_1=[k_1\cos(t)+k_2\text{sen}(t)]\left(\begin{array}{r} 1\\ -1 \end{array}\right) \quad \text{y} \quad w_2=[k_2\cos(t)-k_1\text{sen}(t)]\left(\begin{array}{r} 0\\ 1 \end{array}\right)

Retroalimentación

Pregunta

\pmb{\dot{x}= \left( \begin{array}{rr} -6&5\\ -5&4 \end{array} \right) x}

Respuestas

x=k_1\left(\begin{array}{r} 5\\ 5 \end{array}\right)e^{-t}+k_2\left(\begin{array}{r} 1\\ 2 \end{array}\right)e^{-t}

x=k_1\left(\begin{array}{r} 5\\ 5 \end{array}\right)e^{-t}+k_2\left(\begin{array}{r} 5\\ 5 \end{array}\right)te^{-t}

x=k_1\left(\begin{array}{r} 5\\ 5 \end{array}\right)e^{-t}+k_2\left(\begin{array}{r} 5t+1\\ 5t+2 \end{array}\right)e^{-t}

x=k_1\left(\begin{array}{r} 5\\ 5 \end{array}\right)e^{t}+k_2\left(\begin{array}{r} 5t+1\\ 5t+2 \end{array}\right)e^{t}

Retroalimentación

Pregunta

\pmb{\dot{x}= \left( \begin{array}{rr} 1&-1\\ 1&1 \end{array} \right) x}

Respuestas

x=(w_1+w_2)e^{-t} \quad \text{donde}\quad w_1=[k_1\cos(t)+k_2\text{sen}(t)]\left(\begin{array}{r} 0\\ 1 \end{array}\right) \quad \text{y} \quad w_2=[k_2\text{sen}(t)-k_1\cos(t)]\left(\begin{array}{r} 1\\ 0 \end{array}\right)

x=(w_1+w_2)e^{t} \quad \text{donde}\quad w_1=[k_1\cos(t)+k_2\text{sen}(t)]\left(\begin{array}{r} 0\\ 1 \end{array}\right) \quad \text{y} \quad w_2=[k_2\text{sen}(t)-k_1\cos(t)]\left(\begin{array}{r} 0\\ 1 \end{array}\right)

x=(w_1+w_2)e^{t} \quad \text{donde}\quad w_1=[k_1\cos(t)+k_2\text{sen}(t)]\left(\begin{array}{r} 0\\ 1 \end{array}\right) \quad \text{y} \quad w_2=[k_2\cos(t)-k_1\text{sen}(t)]\left(\begin{array}{r} 1\\ 0 \end{array}\right)

x=(w_1+w_2)e^{t} \quad \text{donde}\quad w_1=[k_1\cos(t)+k_2\text{sen}(t)]\left(\begin{array}{r} 0\\ 1 \end{array}\right) \quad \text{y} \quad w_2=[k_1\cos(t)+k_2\text{sen}(t)]\left(\begin{array}{r} 1\\ 0 \end{array}\right)

Retroalimentación

Pregunta

\pmb{\dot{x}= \left( \begin{array}{rrr} 2&-7&0\\ 5&10&4\\ 0&5&2 \end{array} \right) x}

Respuestas

x=k_1\left(\begin{array}{r} -7\\ 5\\ 5 \end{array}\right)e^{-7t}+k_2\left(\begin{array}{r} -7\\ 3\\ 5 \end{array}\right)e^{-5t}+k_3\left(\begin{array}{r} 4\\ 0\\ 5 \end{array}\right)e^{-2t}

x=k_1\left(\begin{array}{r} -7\\ 5\\ 5 \end{array}\right)e^{7t}+k_2\left(\begin{array}{r} -7\\ 3\\ 5 \end{array}\right)e^{5t}+k_3\left(\begin{array}{r} 4\\ 0\\ 5 \end{array}\right)e^{2t}

x=k\left(\begin{array}{r} -7\\ 5\\ 5 \end{array}\right)e^{7t}+k\left(\begin{array}{r} -7\\ 3\\ 5 \end{array}\right)e^{5t}+k\left(\begin{array}{r} 4\\ 0\\ 5 \end{array}\right)e^{2t}

x=\left(\begin{array}{r} -7\\ 5\\ 5 \end{array}\right)e^{7t}+\left(\begin{array}{r} -7\\ 3\\ 5 \end{array}\right)e^{5t}+\left(\begin{array}{r} 4\\ 0\\ 5 \end{array}\right)e^{2t}

Retroalimentación

Pregunta

\pmb{\dot{x}= \left( \begin{array}{rrr} 1&-1&2\\ -1&1&0\\ -1&0&1 \end{array} \right) x}

Respuestas

x=k\left(\begin{array}{r} 0\\ 2\\ 1 \end{array}\right)e^{t}+(w_1+w_2)e^{t} \quad\text{donde}\quad w_1=[\cos(t)+\text{sen}(t)]\left(\begin{array}{r} 1\\ 0\\ 0 \end{array}\right) \quad \text{y} \quad w_2=[\text{sen}(t)-\cos(t)]\left(\begin{array}{r} 0\\ 1\\ 1 \end{array}\right)

x=k_1\left(\begin{array}{r} 0\\ 2\\ 1 \end{array}\right)e^{t}+(w_1+w_2)e^{t} \quad\text{donde}\quad w_1=[k_1\cos(t)-k_2\text{sen}(t)]\left(\begin{array}{r} 1\\ 0\\ 0 \end{array}\right) \quad \text{y} \quad w_2=[k_2\cos(t)+k_1\text{sen}(t)]\left(\begin{array}{r} 0\\ 1\\ 1 \end{array}\right)

x=k_1\left(\begin{array}{r} 0\\ 2\\ 1 \end{array}\right)e^{t}+(w_1+w_2)e^{t} \quad\text{donde}\quad w_1=[k_2\cos(t)+k_3\text{sen}(t)]\left(\begin{array}{r} 1\\ 0\\ 0 \end{array}\right) \quad \text{y} \quad w_2=[k_3\cos(t)-k_2\text{sen}(t)]\left(\begin{array}{r} 0\\ 1\\ 1 \end{array}\right)

x=k_1\left(\begin{array}{r} 0\\ 2\\ 1 \end{array}\right)e^{t}+(w_1+w_2)e^{-t} \quad\text{donde}\quad w_1=[k_2\cos(t)+k_3\text{sen}(t)]\left(\begin{array}{r} 1\\ 0\\ 0 \end{array}\right) \quad \text{y} \quad w_2=[k_3\text{sen}(t)-k_2\cos(t)]\left(\begin{array}{r} 0\\ 1\\ 1 \end{array}\right)

Retroalimentación

Pregunta

\pmb{\dot{x}= \left( \begin{array}{rr} -6&5\\ -5&4 \end{array} \right) x + \left( \begin{array}{r} 2\\ 1 \end{array} \right)}

Respuestas

x=k_1\left(\begin{array}{r} 5\\ 5 \end{array}\right)e^{-t}+k_2\left(\begin{array}{c} 5t+1\\ 5t+2 \end{array}\right)e^{-t}-\left(\begin{array}{r} 3\\ 4 \end{array}\right)

x=k_1\left(\begin{array}{r} 5\\ 5 \end{array}\right)e^{-t}+k_2\left(\begin{array}{c} 5t+1\\ 5t+2 \end{array}\right)e^{-t}+\left(\begin{array}{r} 3\\ 4 \end{array}\right)

x=k\left(\begin{array}{r} 5\\ 5 \end{array}\right)e^{-t}+k\left(\begin{array}{c} 5\\ 5 \end{array}\right)te^{-t}-\left(\begin{array}{r} 3\\ 4 \end{array}\right)

x=k_1\left(\begin{array}{r} 5\\ 5 \end{array}\right)e^{-t}+k_2\left(\begin{array}{c} 1\\ 2 \end{array}\right)te^{-t}+\left(\begin{array}{r} -3\\ -4 \end{array}\right)

Retroalimentación

Pregunta

\pmb{\dot{x}= \left( \begin{array}{rr} 2&-1\\ 3&-2 \end{array} \right) x + \left(\begin{array}{r} 1\\ -1 \end{array} \right)}

Respuestas

x=k_1\left(\begin{array}{r} 1\\ 3 \end{array}\right)e^{-t}+k_2\left(\begin{array}{r} 1\\ 1 \end{array}\right)e^{t}-\left(\begin{array}{r} 3\\ 5 \end{array}\right)

x=k_1\left(\begin{array}{r} 1\\ 3 \end{array}\right)e^{-t}+k_2\left(\begin{array}{r} 1\\ 1 \end{array}\right)e^{t}+\left(\begin{array}{r} 3\\ 5 \end{array}\right)

x=k\left(\begin{array}{r} 1\\ 3 \end{array}\right)e^{-t}+k\left(\begin{array}{r} 1\\ 1 \end{array}\right)e^{t}+\left(\begin{array}{r} -3\\ -5 \end{array}\right)

x=k_1\left(\begin{array}{r} 1\\ 3 \end{array}\right)e^{t}+k_2\left(\begin{array}{r} 1\\ 1 \end{array}\right)e^{-t}+\left(\begin{array}{r} -3\\ -5 \end{array}\right)

Retroalimentación

Pregunta

\pmb{\dot{x}= \left( \begin{array}{rrr} 0&1&1\\ 1&0&1\\ 1&1&0 \end{array} \right) x + \left(\begin{array}{r} -1\\ 0\\ -1 \end{array} \right)}

Respuestas

x=k_1\left(\begin{array}{r} 1\\ 1\\ 1 \end{array}\right)e^{2t}+k_2\left(\begin{array}{r} -1\\ 0\\ 1 \end{array}\right)e^{-t}+k_3\left(\begin{array}{r} -1\\ 0\\ 1 \end{array}\right)te^{-t}

x=k_1\left(\begin{array}{r} 1\\ 1\\ 1 \end{array}\right)e^{2t}+k_2\left(\begin{array}{r} -1\\ 1\\ 0 \end{array}\right)e^{-t}+k_3\left(\begin{array}{r} -1\\ 1\\ 0 \end{array}\right)te^{-t}

x=k_1\left(\begin{array}{r} 1\\ 1\\ 1 \end{array}\right)e^{2t}+k_2\left(\begin{array}{r} -1\\ 0\\ 1 \end{array}\right)e^{-t}+k_3\left(\begin{array}{r} -1\\ 1\\ 0 \end{array}\right)e^{-t}

x=k\left(\begin{array}{r} 1\\ 1\\ 1 \end{array}\right)e^{2t}+k\left(\begin{array}{r} -1\\ 0\\ 1 \end{array}\right)e^{-t}+k\left(\begin{array}{r} -1\\ 1\\ 0 \end{array}\right)e^{-t}

Retroalimentación

Pregunta

\pmb{\dot{x}= \left(\begin{array}{rr} 1&-2\\ 2&-3 \end{array}\right) x + \left(\begin{array}{r} 1\\ -1 \end{array}\right) \quad \text{considere} \quad x(0)= \left(\begin{array}{r} 6\\ 2 \end{array}\right)}

Respuestas

x= k_1\left(\begin{array}{r} 1\\ 1 \end{array}\right)e^{-t} +k_2 \left(\begin{array}{c} t\\ t-1/2 \end{array}\right)e^{-t} +\left(\begin{array}{r} 5\\ 3 \end{array}\right)

x= k_1\left(\begin{array}{r} 1\\ 1 \end{array}\right)e^{-t} + k_2\left(\begin{array}{c} t\\ t-1/2 \end{array}\right)e^{-t}

x= \left(\begin{array}{r} 1\\ 1 \end{array}\right)e^{-t} +4 \left(\begin{array}{c} t\\ t-1/2 \end{array}\right)e^{-t} +\left(\begin{array}{r} 5\\ 3 \end{array}\right)

x= 6\left(\begin{array}{r} 1\\ 1 \end{array}\right)e^{-t} +8 \left(\begin{array}{c} t\\ t-1/2 \end{array}\right)e^{-t}

Retroalimentación