A system of DE's is given as a matrix, as well as two vectors that are solutions of the system. I'm asked to use the Wronskian to show these solutions are linear independent. I have set up wh at I think the Wronskian is in this case and calculated its determinant to check whether it is not zero. The vectors and Wronskian follow in the next post. Can someone confirm whether I set up the correct Wronskian?

Question

A system of DE's is given as a matrix, as well as two vectors that are solutions of the system. I'm asked to use the Wronskian to show these solutions are linear independent.  I have set up wh at I think the Wronskian is in this case and calculated its determinant to check whether it is not zero. The vectors and Wronskian follow in the next post. Can someone confirm whether I set up the correct Wronskian?

anonymous · Answer

$$x _{1}=\left(\begin{matrix}3e ^{2t} \ 2e ^{2t}\end{matrix}\right)$$
$$x _{2}=\left(\begin{matrix}e ^{-5t} \ 3e ^{-5t}\end{matrix}\right)$$

$$W[x_{1}, x_{2}]=\left[\begin{matrix}3e ^{2t} & e ^{-5t} \ 2e ^{2t}  & 3e ^{-5t}\end{matrix}\right]=9e ^{-3t}-2e ^{-3t}=7e ^{-3t}\neq0$$

anonymous · Answer

yup, the determinant of the matrix shows that x1,x2 are linear independent

anonymous · Answer

It's the first exercise I'm making on this and there's no worked out example in the course. Thanks for verifying for me that I got it!

anonymous · Answer

:)