# Problems with Laplace Transforms

1. May 8, 2014

### faiz4000

1. The problem statement, all variables and given/known data

The coordinates $(x,y)$ of a particle moving along a plane curve at any time t, are given by

$$\frac{dy}{dt} + 2x=\sin 2t,$$
$$\frac{dx}{dt} - 2y=\cos 2t.$$

If at $t=0$, $x=1$ and $y=0$, using Lapace transform show that the particle moves along the curve

$$4x^2+4xy+5y^2=4$$

2. Relevant equations

note: Lowercase letters $x$,$y$ are functions of $t$. Uppercase letters $X$,$Y$ are functions of $s$.

3. The attempt at a solution

Apply Laplace Transform on the given equations

$$\frac{dy}{dt} + 2x=\sin 2t~~~~~~~~(1)$$

Applying LT,

$$sY-y(0)+2X=\frac{2}{s^2+4}$$

$$2X+sY= \frac{2}{s^2+4}~~~~~~~~(2)$$

$$\frac{dx}{dt} - 2y=\cos 2t~~~~~~~~(3)$$

Applying LT,

$$sX-2Y=\frac{s^2+s+4}{s^2+4}~~~~~~~~(4)$$

Now Solving $(2)$ and $(4)$ simultaneously,

$$X=-s^3-s^2-4s-4$$

$$Y=2s^2+8$$

Now I have to apply Inverse Laplace transform to get back $x$ and $y$ but i dont know how to get ILT of a constant........... Also not sure everything I've done till now is right so please help

Last edited by a moderator: May 8, 2014
2. May 8, 2014

### faiz4000

Sorry those fractions didn't come properly

3. May 8, 2014

### faiz4000

Solving Simultaneous Differential Equations using Laplace Transform

1. The problem statement, all variables and given/known data

The coordinates (x,y) of a particle moving along a plane curve at any time t, are given by

$\frac{dy}{dt}$ + 2x=sin2t,
$\frac{dx}{dt}$ - 2y=cos2t

If at t=0, x=1 and y=0, using Lapace transform show that the particle moves along the curve

4x2+4xy+5y2=4

2. Relevant equations

note: Lowercase letters x,y are functions of t. Uppercase letters X,Y are functions of s.

3. The attempt at a solution

Apply Laplace Transform on the given equations

$\frac{dy}{dt}$ + 2x=sin2t -1

Applying LT,

sY-y(0)+2X=$\frac{2}{s2+4}$

2X+sY=$\frac{2}{s2+4}$ -2

$\frac{dx}{dt}$ - 2y=cos2t -3

Applying LT,

sX-2Y=$\frac{s2+s+4}{s2+4}$ -4

Now Solving 2 and 4 simultaneously,

X=-s3-s2-4s-4

Y=2s2+8

Now I have to apply Inverse Laplace transform to get back x and y but i dont know how to get ILT of a constant........... Also not sure everything I've done till now is right so please help

4. May 8, 2014

5. May 8, 2014

### micromass

Staff Emeritus
LaTeX fixed.

6. May 8, 2014

### Staff: Mentor

Hmpf, I left that as an exercise to the reader.

7. May 8, 2014

### Staff: Mentor

Your simultaneous solutions for X and Y are incorrect. Try again.

Chet

8. May 8, 2014

### LCKurtz

Where did the -1 come from?
Where did the -3 come from?

If the -1 and -3 are supposed to be in there, you didn't transform them. And given the equations you have, how did you get those values for X and Y?

9. May 9, 2014

### faiz4000

those were eqn numbers.....eqn 1 and eqn 3.... I had left space but it didn't reflect in the post....I found my mistake.... it was in solving the simultaneous eqns.....you actually get X=(s+1)/(s^2+4)
Y=-s/(s^2+4)

10. May 9, 2014