1. ## Laplace Transform Questionn

Hi

I got a couple of questions that i am having trouble with:

1)Use Laplace transform to solve the following initial value problems:

$\displaystyle y''-2y'+2y=c$

y(0) = 20 and y'(0) = 34

This is what i have done:

$\displaystyle (s^2 Y(s) - sy(0) - y'(0)) - 2(sY(s)-y(0))+2Y(s) = X = \frac{2-cos(t)}{2s}$

$\displaystyle (s^2 Y(s) -20s - 34) - 2(sY(s)-20)+2Y(s) = \frac{2}{(s-1)}+\frac{6}{(s-1)^3}$

$\displaystyle (s^2 - 2s + 2 )Y -20s - 34 +40 = \frac{2}{(s-1)}+\frac{6}{(s-1)^3}$

$\displaystyle (s^2 - 2s + 2 )Y = \frac{2}{(s-1)}+\frac{6}{(s-1)^3} + 20s - 6$

$\displaystyle (s^2 - 2s + 2 )Y = \frac{2(s-1)^2}{(s-1)}+\frac{6}{(s-1)^3} + 20s - 6$

$\displaystyle (s^2 - 2s + 2 )Y = \frac{6+ 2(s-1)^2}{(s-1)^3} + 20s - 6$

$\displaystyle Y = \frac{8+2s^2-4s}{(s-1)^3 (s^2 - 2s + 2 )} + \frac{20s - 6}{(s^2 - 2s + 2 )}$

What should i do next??

2) Use Laplace transform to solve each system of differential equations
$\displaystyle \frac{dx}{dt} = -y+cos(t)$

$\displaystyle \frac{dy}{dt} = x+1$

given x(0) = 1 y(0) = 0

This is what i have done

$\displaystyle sX - 1 = -Y + cos(t)$
$\displaystyle sY = X + 1$

$\displaystyle sX + Y = 1+ cos(t)$
$\displaystyle sY - X = 1$

Matrix Form to solve X and Y

| s 1 |
| s -1 |

Det= -2s

| 1+cos(t) 1 |
| 1 -1 |

$\displaystyle Det_x = -2-cos(t)$

| s 1+cos(t) |
| s 1 |

$\displaystyle Det_y = -scos(t)$

$\displaystyle X = \frac{2-cos(t)}{2s}$
$\displaystyle Y = \frac{cos(t)}{2}$
what is wrong?

P.S

2. Originally Posted by Paymemoney

2) Use Laplace transform to solve each system of differential equations
$\displaystyle \frac{dx}{dt} = -y+cos(t)$

$\displaystyle \frac{dy}{dt} = x+1$

given x(0) = 1 y(0) = 0

This is what i have done

$\displaystyle sX - 1 = -Y + cos(t)$
$\displaystyle sY = X + 1$

$\displaystyle sX + Y = 1+ cos(t)$
$\displaystyle sY - X = 1$

Matrix Form to solve X and Y

| s 1 |
| s -1 |

Det= -2s

| 1+cos(t) 1 |
| 1 -1 |

$\displaystyle Det_x = -2-cos(t)$

| s 1+cos(t) |
| s 1 |

$\displaystyle Det_y = -scos(t)$

$\displaystyle X = \frac{2-cos(t)}{2s}$
$\displaystyle Y = \frac{cos(t)}{2}$
what is wrong?

P.S
You have not taken the LT of $\displaystyle \cos(t)$

CB