Maths for economists2

Maths for economists Problem Set III

due 11/19/15

Total is 20 points but 24 points are possible. Show your work.

1 First-order differential equations (1 point each)

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 (1 point each)

Solve the following and study the behavior of xt when t → ∞. a 4xt+1 − 2xt = 4, with x0 = 2 b 4xt+1 + 2xt = 4, with x0 = 2 c 2xt+1 − 2xt = 7, with x0 = 0 d 2xt+1 + 2xt = 3, with x0 = 0

3 Second-order differential and difference equations (1 point each)

Solve the following and study the behavior of xt 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 xt+2 + 4xt+1 + 3xt = 10, with x0 = 0 and x1 = 5 e xt+2 + 4xt+1 + 4xt = 10, with x0 = 0 and x1 = 5

1

4 Systems of differential and difference equations (3.5 points each)

Solve the following systems:

a Y ′ = AY + K with A = (

2 2 3 1

) and K =

( 2 2

) and Y (0) =

( 4 1

)

b Y t+1 = AY t + K with A = (

2 2 3 1

) and K =

( 2 2

) and Y 0 =

( 4 1

)

5 Diagrammatic analysis

Consider the system of equations 4 a.

a Perform a diagrammatic analysis. (3 points)

b Will the system converge? (1 point)

2