Concrete Solutions
Question
Solution
Short
Video
\(\LaTeX\)
No explanation / solution video to this exercise has yet been created.
Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Exercise:
Find the concrete solutions of two-dimensional linear systems for the given eigenvalues eigenvectors and initial conditions. abcliste abc lambda_ - lambda_ - bf v_ leftmatrix matrixright bf v_ leftmatrix matrixright leftmatrixx_ y_ matrixright leftmatrix - matrixright abc lambda_ lambda_ - bf v_ leftmatrix matrixright bf v_ leftmatrix matrixright leftmatrixx_ y_ matrixright leftmatrix matrixright abc lambda_ i lambda_ -i bf v_ leftmatrix+i matrixright bf v_ leftmatrix-i matrixright leftmatrixx_ y_ matrixright leftmatrix matrixright abcliste
Solution:
The solutions can be written as leftmatrixx y matrixright a_ bf v_ e^lambda_ t+a_ bf v_ e^lambda t The coefficients a_ and a_ are given by the initial conditions: leftmatrixx_ y_ matrixright a_ bf v_ +a_ bf v_ abcliste abc a_+a_ - a_ Longrightarrow a_ a_ - The solution is xt result e^-t - e^- t yt result- e^- t abc a_+a_ a_+ a_ Longrightarrow a_ frac a_ -frac The solution is xt resultfrac e^t-frac e^-t yt resultfrac e^t-frac e^-t abc +ia_+-ia_ a_+ a_ Longrightarrow a_ -a_+frac Longrightarrow a_ +i+-ifrac-a_ i a_+frac-i Longrightarrow a_ frac+ifracifrac-i a_ frac+i The solution is xt frac-i+i e^iomega t + frac+i-ie^-iomega t fraclefte^iomega t+e^-iomega tright+fraclefte^iomega t+e^-iomega tright & quad -fracilefte^iomega t-e^-iomega tright+fracilefte^iomega t-e^-omega tright frac cosomega t-fraci i sinomega t resultcosomega t+sinomega t yt frac-i e^iomega t+frac+i e^-iomega t fraclefte^iomega t+e^-iomega tright & quad -fracilefte^iomega t-e^-iomega tright frac cosomega t-fraci i sinomega t resultcosomega t+sinomega t abcliste
Find the concrete solutions of two-dimensional linear systems for the given eigenvalues eigenvectors and initial conditions. abcliste abc lambda_ - lambda_ - bf v_ leftmatrix matrixright bf v_ leftmatrix matrixright leftmatrixx_ y_ matrixright leftmatrix - matrixright abc lambda_ lambda_ - bf v_ leftmatrix matrixright bf v_ leftmatrix matrixright leftmatrixx_ y_ matrixright leftmatrix matrixright abc lambda_ i lambda_ -i bf v_ leftmatrix+i matrixright bf v_ leftmatrix-i matrixright leftmatrixx_ y_ matrixright leftmatrix matrixright abcliste
Solution:
The solutions can be written as leftmatrixx y matrixright a_ bf v_ e^lambda_ t+a_ bf v_ e^lambda t The coefficients a_ and a_ are given by the initial conditions: leftmatrixx_ y_ matrixright a_ bf v_ +a_ bf v_ abcliste abc a_+a_ - a_ Longrightarrow a_ a_ - The solution is xt result e^-t - e^- t yt result- e^- t abc a_+a_ a_+ a_ Longrightarrow a_ frac a_ -frac The solution is xt resultfrac e^t-frac e^-t yt resultfrac e^t-frac e^-t abc +ia_+-ia_ a_+ a_ Longrightarrow a_ -a_+frac Longrightarrow a_ +i+-ifrac-a_ i a_+frac-i Longrightarrow a_ frac+ifracifrac-i a_ frac+i The solution is xt frac-i+i e^iomega t + frac+i-ie^-iomega t fraclefte^iomega t+e^-iomega tright+fraclefte^iomega t+e^-iomega tright & quad -fracilefte^iomega t-e^-iomega tright+fracilefte^iomega t-e^-omega tright frac cosomega t-fraci i sinomega t resultcosomega t+sinomega t yt frac-i e^iomega t+frac+i e^-iomega t fraclefte^iomega t+e^-iomega tright & quad -fracilefte^iomega t-e^-iomega tright frac cosomega t-fraci i sinomega t resultcosomega t+sinomega t abcliste