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4.11. Derive the Laplace transform for the following function using the definition of the Laplace transform (Equation 4.2.1). f(t)=a1a21(ea2tea1t) 4.14. Determine the Laplace transform of f(t) for the following functions using Table 4.1. a. f(t)=et+eatsint b. f(t)=dtdg(t)e2tcos5t c. f(t)=0tg(t)dtt3e3t d. f(t)=7g(t10)+sin10t 4.15. Apply the initial- and final-value theorems to f(t) for the following functions. a. f(t)=eetsint b. f(t)=t2e5t+sin5t c. f(t)=e2tsin2te2t d. f(t)=sint+cost 4.21 Determine y(t) for each of the following differential equations, assuming that y(0)=(dtdy)m=0=0 a. dt2d2y+5dtdy+6y=2 b. dt2d2y+3dtdy=5 4.11. Derive the Laplace transform for the following function using the definition of the Laplace transform (Equation 4.2.1). f(t)=a1a21(ea2tea1t) 4.14. Determine the Laplace transform of f(t) for the following functions using Table 4.1. a. f(t)=et+eatsint b. f(t)=dtdg(t)e2tcos5t c. f(t)=0tg(t)dtt3e3t d. f(t)=7g(t10)+sin10t 4.15. Apply the initial- and final-value theorems to f(t) for the following functions. a. f(t)=eetsint b. f(t)=t2e5t+sin5t c. f(t)=e2tsin2te2t d. f(t)=sint+cost 4.21 Determine y(t) for each of the following differential equations, assuming that y(0)=(dtdy)m=0=0 a. dt2d2y+5dtdy+6y=2 b. dt2d2y+3dtdy=5