Laplace differential equation

ammar · 2013-12-17 16:38:55

I do not know this is home work

bobbym · 2013-12-17 18:45:43

The best I could do was.

y''(t) - (1-t) y'(t) + y(t) = sin(2 t)

With y(0) = y'(0) = 5

s^2 Y(s) - s y(0) - y'(0) - (s*Y(s) - y(0)) + (-Y(s) + s*Y'(s)) + Y(s) = L(sin(2 t))

Which when solved for Y'(s) just yields another DE.

Y'(s) = ( 2 + 20 s + 5 s^3 + 4 s Y(s) - 4 s^2 Y(s) + s^3 Y(s)- s^4 Y(s))/( s (4 + s^2))

This one I could not solve by hand and when I used Mathematica it again comes up with the Error Function, just as expected.

ammar · 2013-12-17 21:00:12

What you think of using laplace inverse for this equation ?
(We usually take Laplace inverse for last equation and convert it to f(t) function )

bobbym · 2013-12-17 22:24:12

Hi;

Yes, that is what you do when it works, here it is not working. I think the reason for that is because the DE does not have a closed form.

ammar · 2013-12-17 22:25:37

I will leave it at this level

bobbym · 2013-12-17 22:43:31

I can show you an example or two where it works out or you can try the SE and perhaps they can help you. They can be a bit brusque but if you find the right one...

ammar · 2013-12-17 22:57:20

Yes I would be grateful

bobbym · 2013-12-17 23:14:52

Okay, hold on and I will post an example.

Solve:

with

Making all the substitutions:

putting in the initial conditions and solving for Y(s).

Do you know how to take the inverse Laplace of the RHS?

ammar · 2013-12-17 23:27:07

Yes, by partial fraction

bobbym · 2013-12-17 23:27:59

And what did you get?

ammar · 2013-12-17 23:34:18

Y(t)=5/4-(5/4)exp(-4t)

bobbym · 2013-12-17 23:38:34

Hi;

That is close but not quite correct. The correct answer is.

ammar · 2013-12-17 23:46:09

I think 5/4(e(-4t) because by parital .. A=-B and A=5/4

bobbym · 2013-12-17 23:51:37

Hi;

That is not correct,

Take the Inverse Laplace of that.

ammar · 2013-12-18 00:18:45

y(t)=5/4-exp(-4t)/4 {{{that's right}}}

ammar · 2013-12-18 01:46:41

But this is easy one but the others .......

bobbym · 2013-12-18 02:42:58

It is easy. Whomever picked the other two probably wanted you to leave them in some form which I can only guess at.

ammar · 2013-12-18 03:06:54

Yes, I think so

bobbym · 2013-12-18 03:07:52

Do you have an example?

ammar · 2013-12-18 03:22:23

Yes

y'+y=1 where y(0)=0

y'+2y=1+exp(t) where y(0)=3

t
y+∫ y(t)dt=1
0

y''-y'-2y=3cos(2t)-11sin(3t) where y(0)=0, y'(0)=6

bobbym · 2013-12-18 03:28:00

Okay, hold on I will do some.

Substituting for y(0)=0.

That is the answer.

ammar · 2013-12-18 04:13:36

correct answer , What program use to solve ?

bobbym · 2013-12-18 04:20:26

I did it by hand using the method of Laplace transforms.

Mathematica did the checking that the answer was right but Wolfram alpha will do it too.

ammar · 2013-12-18 04:32:37

we soon have exam and I still good enough with laplace and other subjects .

thanks for helping me and communicate with me

zetafunc. · 2013-12-18 04:33:07

ammar wrote:

correct answer , What program use to solve ?

No program necessary -- the Laplace transform of y' is a common identity, and the transforms of 1/s and 1/(s+a) are well-known (you will find them in any table of transforms).

Math Is Fun Forum

#26 2013-12-17 16:38:55

Re: Laplace differential equation

#27 2013-12-17 18:45:43

Re: Laplace differential equation

#28 2013-12-17 21:00:12

Re: Laplace differential equation

#29 2013-12-17 22:24:12

Re: Laplace differential equation

#30 2013-12-17 22:25:37

Re: Laplace differential equation

#31 2013-12-17 22:43:31

Re: Laplace differential equation

#32 2013-12-17 22:57:20

Re: Laplace differential equation

#33 2013-12-17 23:14:52

Re: Laplace differential equation

#34 2013-12-17 23:27:07

Re: Laplace differential equation

#35 2013-12-17 23:27:59

Re: Laplace differential equation

#36 2013-12-17 23:34:18

Re: Laplace differential equation

#37 2013-12-17 23:38:34

Re: Laplace differential equation

#38 2013-12-17 23:46:09

Re: Laplace differential equation

#39 2013-12-17 23:51:37

Re: Laplace differential equation

#40 2013-12-18 00:18:45

Re: Laplace differential equation

#41 2013-12-18 01:46:41

Re: Laplace differential equation

#42 2013-12-18 02:42:58

Re: Laplace differential equation

#43 2013-12-18 03:06:54

Re: Laplace differential equation

#44 2013-12-18 03:07:52

Re: Laplace differential equation

#45 2013-12-18 03:22:23

Re: Laplace differential equation

#46 2013-12-18 03:28:00

Re: Laplace differential equation

#47 2013-12-18 04:13:36

Re: Laplace differential equation

#48 2013-12-18 04:20:26

Re: Laplace differential equation

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Re: Laplace differential equation

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Re: Laplace differential equation

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