ODELS2

From Example Problems
Jump to: navigation, search

Find a fundamental matrix and the Wronskian for the following linear system: {\begin{cases}x_{1}'=x_{2}\\x_{2}'=x_{1}\end{cases}}\,.

The matrix of coefficients A\, is {\begin{bmatrix}0&1\\1&0\end{bmatrix}}\,.

The eigenvalues are \lambda =\pm 1\,.

For \lambda =1\, an eigenvector is u_{1}={\begin{bmatrix}1\\1\end{bmatrix}}\,.

For \lambda =-1\, an eignvector is u_{{-1}}={\begin{bmatrix}-1\\1\end{bmatrix}}\,.

Therefore two linearly independent solutions are:

x^{1}=e^{t}{\begin{bmatrix}1\\1\end{bmatrix}}={\begin{bmatrix}e^{t}\\e^{t}\end{bmatrix}}\,

x^{2}=e^{{-t}}{\begin{bmatrix}-1\\1\end{bmatrix}}={\begin{bmatrix}-e^{{-t}}\\e^{{-t}}\end{bmatrix}}\,

So a fundamental matrix is X(t)={\begin{bmatrix}e^{t}&-e^{{-t}}\\e^{t}&e^{{-t}}\end{bmatrix}}\,.

The Wronskian is the determinant of this matrix after evaluation at t=0\,.

W=1+1=2\,


Main Page : Ordinary Differential Equations : Linear Systems