First-order differential equations solve the following and

Question

1. First-order differential equations

Solve the following and study the behavior of x(t) when t → ∞.

a. 2x'(t)+8x(t) = 6, with x(0) = 10

b. x'(t)+5x(t)/t = t, with x(1) = 1

c. x'(t)+ x(t) = t, with x(1) = 10

d. x'(t)+ t x(t) = t, with x(0) = 1

2. First-order difference equations

Solve the following and study the behavior of x_t when t → ∞.

a. 4x_t+1 -2x_t = 4, with x₀ = 2

b. 4x_t+1 +2x_t = 4, with x₀ = 2

c. 2x_t+1 -2x_t = 7, with x₀ = 0

d 2x_t+1 +2x_t = 3, with x₀ = 0

3. Second-order differential and difference equations

Solve the following and study the behavior of x_t when t → ∞.

a. 2x''(t)+8x(t) = 0, with x(0) = 1 and x'(0) = 0

b. x''(t)+5x'(t)+6x(t) = 1, with x(0) = 1 and x'(0) = 0

c. x''(t)+2x'(t)+ x(t) = 1, with x(0) = 1 and x'(0) = 1

d. x_t+2 +4x_t+1 +3x_t = 10, with x₀ = 0 and x₁ = 5

e. x_t+2 +4x_t+1 +4x_t = 10, with x₀ = 0 and x₁ = 5

4. Systems of differential and difference equations

Solve the following systems:

5. Diagrammatic analysis

Consider the system of equations 4 a.   

a. Perform a diagrammatic analysis.

b. Will the system converge?