2) Solve the following initial value proḅlem using the Laplace transform. a) y′′+2y′−3y=4y(0)=0y′(0)=4 b) y′′−4y′+3y=e−3ty(0)=0y′(0)=1 c) y′′−3y′+2y=ty(0)=y′(0)=0 d) y′′+7y′+12y=2+e−2ty(0)=0y′(0)=0 e) y′′+10y=δ(t−2π)y(0)=0y′(0)=1 g) y′′+9y=3δ(t−4)+9u7(t)y(0)=0y′(0)=7

a) Applying the Laplace transform to both sides of the differential equation and using the fact that...

Resuelto 2) Solve the following initial value proḅlem using

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Pregunta: 2) Solve the following initial value proḅlem using the Laplace transform. a) y′′+2y′−3y=4y(0)=0y′(0)=4 b) y′′−4y′+3y=e−3ty(0)=0y′(0)=1 c) y′′−3y′+2y=ty(0)=y′(0)=0 d) y′′+7y′+12y=2+e−2ty(0)=0y′(0)=0 e) y′′+10y=δ(t−2π)y(0)=0y′(0)=1 g) y′′+9y=3δ(t−4)+9u7(t)y(0)=0y′(0)=7

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The first step involves applying the Laplace transform to both sides of the differential equation , and exploiting the properties of the Laplace transform such that and .
Paso 1
a) Applying the Laplace transform to both sides of the differential equation and using the fact that...
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Paso 2
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2) Solve the following initial value proḅlem using the Laplace transform. a)

y^{''} + 2 y^{'} - 3 y = 4 y (0) = 0 y^{'} (0) = 4

y^{''} - 4 y^{'} + 3 y = e^{- 3 t} y (0) = 0 y^{'} (0) = 1

y^{''} - 3 y^{'} + 2 y = t y (0) = y^{'} (0) = 0

y^{''} + 7 y^{'} + 12 y = 2 + e^{- 2 t} y (0) = 0 y^{'} (0) = 0

y^{''} + 10 y = δ (t - 2 π) y (0) = 0 y^{'} (0) = 1

y^{''} + 9 y = 3 δ (t - 4) + 9 u_{7} (t) y (0) = 0 y^{'} (0) = 7