# Questions in Section Differential Equations: page 2

Use the Laplace transform to solve the differential equation: y''+y=\delta(t-\pi), y(0)=0, y'(0)=0.

Use the Laplace transform to solve the differential equation: y''-y'-2y=u_3(t)(t-3), y(0)=0, y'(0)=0.

Use the Laplace transform to solve the differential equation: y''+4y=u_1(t), y(0)=0, y'(0)=0.

Find the inverse Laplace transform of the function F(s): F(s)=(e^(-s))/(s^2+1).

Write the following piecewise defined function as a step function: f(t)={(t if 0<=t<1),(t^2 if 1<=t<2), (0 if 2<=t):}.

Use the Laplace transform to solve the differential equation: y''+4y'+3y=e^t, y(0)=0, y'(0)=0.

Use the Laplace transform to solve the differential equation: y''-2y'-3y=0, y(0)=0, y'(0)=1.

Find the inverse Laplace transform of the function F(s): F(s)=1/(s^2-3s+2).

Use the definition of Laplace transform to find L(f(t)) of piecewise defined function: f(t)={(1 if 0<=t<1),(2 if 1<=t<2),(0 if 2<=t):}.

A stream, which is polluted with insecticide at concentration 8 g/m^3, flows at a rate of 20 (m^3)/(day) into a pond of volume 1000 m^3. At the same time, water from the pond is flowing into the sea at rate 20 (m^3)/(day). The initial insecticide concentration in the pond is 3 g/m^3.