If y(0)=2 and y'(0)=2k, then:
ℒ{y"}=s2Y(s)-sy(0)-y'(0)=s2Y(s)-2s-2k; ℒ{y'}=sY(s)-y(0)=sY(s)-2.
For Y(s), simply write Y.
ℒ{y"+ky'-2k2y}=s2Y-2s-2k+skY-2k-2k2Y=0.
Y can be factored out:
Y(s2+sk-2k2)=2s+4k,
Y(s+2k)(s-k)=2s+4k=2(s+2k),
Y(s-k)=2, Y=2/(s-k). Apply inverse Laplace:
2ℒ-1{1/(s-k)}=2ekt is the solution.