Need help with solving laplace transform question

[pokin]

I’ve attached a photo to the question and my working.
The answer is supposedly suppose to be:
(s+3)L{V}-6-26/(s^2+4)

 

[HallsofIvy]

You want find the Laplace Transform of $\displaystyle \frac{dv}{dt}+ 3v- 13sin(2t)$
Yes, the Laplace Transform is "linear" so you can take the Laplace transform of each part:
Letting the Laplace Transform of v be "V" the Laplace transform of $\displaystyle \frac{dv}{dt}$ is $\displaystyle sV- v(0)= sV- 6$.

The Laplace Transform of $\displaystyle -13 sin(2t)$ is (either using a table of Laplace transforms or by actually doing the integration by parts twice) $\displaystyle -13\frac{2}{s^2+ 4}= \frac{-26}{s^2+ 4}$.

So the Laplace Transform of $\displaystyle \frac{dv}{dt}+ 3v- 13sin(2t)$ is
$\displaystyle sV- 6+ 3V- \frac{26}{s^2+ 4}$.

In your solution, you appear to have $\displaystyle L(3v)= 3L(v)= \frac{3}{s^2}$. Since, in your notation, you are using "V" to represent the Laplace transform of v, the Laplace transform of 3v is just 3V. I suspect you used the Laplace transform of independent variable, "t", not the dependent variable, "v".