I'm having trouble with this rather simple problem while revising integration:
a) Differentiate and with respect to .
b) Hence show that
and
c) Use the two equations from b to determine
Thank you for your time.
First of all, you have a sign error in $\displaystyle \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \left[ \mathrm{e}^{-3x}\cos{ \left( 2x \right) } \right] = -3\mathrm{e}^{-3x}\cos{ \left( 2x \right) } - 2\mathrm{e}^{-3x}\sin{ \left( 2x \right) } = -\mathrm{e}^{-3x} \left[ 3\cos{ \left( 2x \right) } + 2 \sin{ \left( 2x \right) } \right] \end{align*}$. This might have affected your chances to get further...
The answer for the second derivative of part (a) is missing a negative sign(-) in the front.
Part (b) follows from part (a) from the difference rule of integration.
Part (c) can be found by multiplying both equations from (b) by a constant so as to eliminate the when adding both equations together.