# Math Help - Laplace Transform For First Order ODE

1. ## Laplace Transform For First Order ODE

This is literally the first day ive ever encountered laplace transformation so I'm gonna apologise in advance if the answer to this question is blantantly obvious. I am currently trying to solve the following equation using laplace transformation:

$dy/dt + 3y = e^{-5t}$
Subject to the initial condition that y(0) = 1

The laplace transform I have as:

$[sX(s) - y(0)] + 3X(s) = 1/(s + 5)$

In all the previous examples I have thus far completed the initial condition has been y(0) = 0, and therefore the y(0) term in the laplace equation can be disregarded leaving:

$X(s) = 1/(s + 3)(s + 5)$

Which is in my laplace transform tables under:

$X(s) = k/(s + a)(s + B)$

Where k, a and B are constants

However if the intitial condition is y(0) = 1, the equation now becomes:

$X(s) = 1/(s + 3)(s + 5) + 1$

I now have the issue that the equation no longer fits my laplace table equation and I cannot locate an inverse laplace transform simply for a constant. Does anyone know how this equation can be rearranged back in terms of x(t)? Thanks

2. Originally Posted by MathsDude69
This is literally the first day ive ever encountered laplace transformation so I'm gonna apologise in advance if the answer to this question is blantantly obvious. I am currently trying to solve the following equation using laplace transformation:

$dy/dt + 3y = e^-5t$
Subject to the initial condition that y(0) = 1

The laplace transform I have as:

$[sX(s) - y(0)] + 3X(s) = 1/(s + 5)$

In all the previous examples I have thus far completed the initial condition has been y(0) = 0, and therefore the y(0) term in the laplace equation can be disregarded leaving:

$X(s) = 1/(s + 3)(s + 5)$

Which is in my laplace transform tables under:

$X(s) = k/(s + a)(s + B)$

Where k, a and B are constants

However if the intitial condition is y(0) = 1, the equation now becomes:

$X(s) = 1/(s + 3)(s + 5) + 1$

I now have the issue that the equation no longer fits my laplace table equation and I cannot locate an inverse laplace transform simply for a constant. Does anyone know how this equation can be rearranged back in terms of x(t)? Thanks

Use partial fractions to obtain two fractions

3. Hey thanks for your help. I have broken the solution down into 2 fractions:

$X(s) = 0.5/(s+3) -0.5/(s+5) + 1$

Again these 2 smaller fractions fit into the laplace tables under:

$X(s) = k/(s + a)$
Where k and a are constants.

However im still struggling with the +1 part. Ive read somewhere that the inverse laplace for 1 is the dirac delta function, but that is making the differential equation more obtruse than when I started :-)

Is there something I am missing with this?

4. Originally Posted by MathsDude69
Hey thanks for your help. I have broken the solution down into 2 fractions:

$X(s) = 0.5/(s+3) -0.5/(s+5) + 1$

Again these 2 smaller fractions fit into the laplace tables under:

$X(s) = k/(s + a)$
Where k and a are constants.

However im still struggling with the +1 part. Ive read somewhere that the inverse laplace for 1 is the dirac delta function, but that is making the differential equation more obtruse than when I started :-)

Is there something I am missing with this?
Yes. Your solution for X is wrong. It should be $X(s) = \frac{1}{(s + 5)(s + 3)} + \frac{1}{s + 3}$.