Problem 2 Night Sections

[Victor Ivrii]

7.4 p. 395 \#6
Consider the vectors $\mathbf{x}^{(1)}(t) = \begin{pmatrix}t\\1\end{pmatrix}$ and  $\mathbf{x}^{(2)}(t) = \begin{pmatrix}t^2\\2t\end{pmatrix}$.


(a) Compute the Wronskian of $\mathbf{x}^{(1)}$ and $\mathbf{x}^{(2)}$.

(b) In what intervals are $\mathbf{x}^{(1)}$ and $\mathbf{x}^{(2)}$ linearly independent?

(c) What conclusion can be drawn about the coefficients in the system of homogeneous differential equations satisfied by $\mathbf{x}^{(1)}$ and $\mathbf{x}^{(2)}$?

[Yangming Cai]

 
 
(a). $W(\mathbf{x}^{(1)},\mathbf{x}^{(2)})=t\cdot 2t-1cdot t^2=t^2$;

2. When $t=0$ we have $W=0$;  then $\mathbf{x}^{(1)}(0)$ and $\mathbf{x}^{(2)}(0)$  are linearly dependent, so $\mathbf{x}^{(1)}$ and $\mathbf{x}^{(2)}$ are linearly independant on intervals where $t\ne 0$;

3. The coefficients of the ODE are discontinous at x=0. If $\mathbf{x}$ satisfies this system $\mathbf{x}'+A\mathbf{x}=0$ then $A$ must be singular at $t=0$.