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`.

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