4. (10pts) Consider the damped forced harmonic oscillator with mass 1 kg, damping coefficient 2, spring constant 3, and external force in the form of an instantaneous hammer strike (Section 6.4) at time t = 4 seconds. The mass is initially displaced 2 meters in the positive direction and an initial velocity of 1 m/s is applied. Model this situation with an initial value problem and solve it using the method of Laplace transforms.

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.

a) Verify that y1 and y2 are fundamental solutions of the given homogenous second-order linear differential equation

b) find the general solution for the given differential equation

c) find a particular solution that satisfies the specified initial conditions for the given differential equation

y'' - y = 0 y1 = e^x, y2 = e^-x : y(0) = 0, y'(0) = 5

Determine if the following subsets are subspaces:
1. The set of grade 7 polynomials
2. The set of polynomials of degree 5 such that P (0) = 0
3. The set of continuous functions such that f (0) = 2

Expand in Fourier series:

f(x) = x|x|, -L<x<L, L>0

f(x) = cosx(sinx)^2 , -pi<x<pi

f(x) = (sinx)^3, -pi<x<pi

Expand in Fourier series:

Expand in fourier sine and fourier cosine series of: f(x) = x(L-x),    0<x<L

Expand in fourier cosine series: f(x) = sinx, 0<x<pi

Expand in fourier series f(x) = 2pi*x-x^2, 0<x<2pi, assuming that f is periodic of period 2pi, that is, f(x+2pi)=f(x)

write the theory and formulas for solving the systems of equations using the Laplace transform. Must contain bibliography

NOTE- If it is true, you need to prove it and If it is false, give a counterexample

f : [a, b] → R is continuous and in the open interval (a,b) differentiable.

a) If f(a) ≥ f(b), then exists a ξ ∈ (a,b) with f′(ξ) ≤ 0.(TRUE or FALSE?)

b) If f is reversable, then f −1 differentiable. (TRUE or FALSE?)
c) If f ′ is limited, then f is lipschitz. (TRUE or FALSE?)

Logic. Identify the form of the following statements:

1. ~ (∃x)((Gx ⊃ Hx) ⋅ (Jx ∨ (Hx ⋅ Kx)))
2. (x)((Gx ⊃ Hx) ⊃ (Jx ∨ Kx))
3. (x)(~ Gx ⊃ Hx) ⊃ (x)(Jx)

2. Consider the plane curve r(t) = <2cos(t),3sin(t)>. Parameterize the osculating circle at t=0. Sketch both the curve and the osculating circle

A. Find all the solutions for the congruence ax ≡ d (mod b) a=123, b=456, d=3

using theorem 11.10 (First Isomorphism Theorem), Show that the set of positive real numbers with multiplication is isomorphic to the set of real numbers with addition.

Theorem 11.10 First Isomorphism Theorem. If ψ : G → H is a group homomorphism with K = kerψ, then K is normal in G. Let ϕ : G → G/K be the canonical homomorphism. Then there exists a unique isomorphism η : G/K → ψ(G) such that ψ = ηϕ.