Given differntial equation is,

\(2y"+4y'+17y=3\cos(2t)\)

\(y(0)=y'(0)=0\)

\(L\left[y"\right]=s^2L\left\{y(t)\right\}-sy(0)-y'(0)\)

\(L\left[y'\right]=sL\left\{y(t)\right\}-y(0)\)

\(L\left[\cos(at)\right]=\frac{s}{s^2+a^2}\)

Taking Laplace transform of equation (1),

\(2L\left[y"\right]+4L\left[y'\right]+17L\left[y\right]=3L\left[\cos(2t)\right]\)

\(s^2L\left\{y(t)\right\}-sy(0)-y'(0)+4\left[sL\left\{y(t)\right\}-y(0)\right]+17L\left\{y(t)\right\}=3L\left\{\cos(2t)\right\}\)

\(s^2Ly(t)+4sLy(t)+17Ly(t)=3x\frac{s}{s^2+4}\)

\((s^2+4s+17)L\left\{y(t)\right\}=\frac{3s}{s^2+4}\)

\(a) \therefore L\left\{y(t)\right\}=\frac{3s}{s^2+4} \times \frac{1}{s^2+4s+17}\)

\(b) y(t)=\int_0^t \left[3\cos(3w)\right] \times \left[e^{-2t} \cdot \sin \sqrt{13}w\right] dw\)

This is required Laplace transform.