Question

Show by wronskian that the following functions are linearly independent over all reals: e^axcosbx,e^axsinbx

Answer 1

$$f_1(x)=e^{ax}\cos bx, f_2(x)=e^{ax}\sin bx$$

$$(e^{ax}\cos bx)'=ae^{ax}\cos bx-be^{ax}\sin bx$$

$$(e^{ax}\sin bx)'=ae^{ax}\sin bx+be^{ax}\cos bx$$

$$W(f_1, f_2)=\begin{vmatrix} f_1 & f_2 \\ f_1' & f_2' \end{vmatrix}$$

$$=\begin{vmatrix} e^{ax}\cos bx & e^{ax}\sin bx \\ a e^{ax}\cos bx-be^{ax}\sin bx & ae^{ax}\sin bx+be^{ax}\cos bx \end{vmatrix}$$

$$=ae^{2ax}\sin bx\cos bx+be^{2ax}\cos^2 bx$$

$$-ae^{2ax}\sin bx\cos bx+be^{2ax}\sin^2 bx$$

$$=be^{2ax}\not =0, b\not=0, x\in \R$$

Since $W(f_1, f_2)\not=0, b\not=0, x\in \R,$ then  the given functions are linearly independent over all reals if $b\not=0.$