Question

1. Consider the undamped forced harmonic oscillator with mass 1 kg, damping coefficient 0, spring constant 4, and external force h(t) = 3cos(t). The mass is initially at rest in the equilibrium position. You must understand that you can model this as: y’’ = -4y +3cost; y(0) = 0; y’(0) = 0.
   1. (5pts) Using the method of Laplace transforms, solve this initial value problem.
   2. Check your solution solves the IVP.
      1. (4pts) Be sure to check that your solution satisfies both the differential equation and
      2. (1pt) the two initial conditions.

Answer 1

mass is kg

.

there is no damping so damping constant is

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spring constant is

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external force is

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DE is given by

take Laplace

here spring is in equilibrium position so y(0)=0

and the initial velocity is zero so y'(0)=0

take partial fraction

....................(1)

compare coefficient both sides

put constants in equation1

take inverse Laplace

.

.

.

check our solution

put all values in DE

here both sides are equal so our solution is correct

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