Solving an Initial value problem using Laplace transform

[Rubik]

Homework Statement

Solve the Initial value problem using Laplace transform
$\ddot{y}$ +2y = 0, y₀ = C₁, $\dot{y}$ = C₂

Homework Equations

[s² - sy(0) - y'(0)] + a[sY - y(0)] + bY

The Attempt at a Solution

s²Y - sy(0) - y'(0) + 2y = 0
s²Y + 2Y = sy(0) + y'(0)
(s² + 2)Y = s(C₁) + (C₂)
Y = (s(C₁))/[s² + 2] + (C₂)/[s² + 2]

And that is as far as I can get as I am unsure what to do now?

 

[rock.freak667]

Well the laplace transform of what function gives the form s/(s²+k²)?

Similarly what function's transform give k/(s²+k²)?



(Hint:Think trig functions)

 

[Rubik]

Oh so I use cos$\omega$t

 

[Rubik]

Actually I am confused does that mean for the first bit my answer is cos($\sqrt{2}$ t) what happens to the C₁?

 

[rock.freak667]

Oh so I use cos$\omega$t

Right so cosωt would give ω/(s²+ω²), so comparing this to C₁* s/(s²+2) what is ω ?

Actually I am confused does that mean for the first bit my answer is cos($\sqrt{2}$ t) what happens to the C₁?

The C₁ should be there, even if your book says it isn't there, it should be there still.

 

[Rubik]

so is my answer just C₁cos($\sqrt{2}$ t) otherwise I am really confused..

 

[Rubik]

Right so cosωt would give ω/(s²+ω²), so comparing this to C₁* s/(s²+2) what is ω ?



My book says it is s/(s² + $\omega$²) and not a numerator of $\omega$

 

[rock.freak667]

so is my answer just C₁cos($\sqrt{2}$ t) otherwise I am really confused..

No, remember you have two functions.

Right so cosωt would give ω/(s²+ω²), so comparing this to C₁* s/(s²+2) what is ω ?


My book says it is s/(s² + $\omega$²) and not a numerator of $\omega$

Sorry, I wrote the other one.

s/(s²+ω²) compared to s/(s²+2) gives ω as?


If you are confused as to what I am trying to get you to see, look up the laplace transforms for cosine and sine.

 

[Rubik]

so $\omega$² = 2 which implies $\omega$ = $\sqrt{2}$?

 

[vela]

Good. So you have $$Y(s) = C_1 \frac{s}{s^2+\omega^2} + (C_2/\omega)\frac{\omega}{s^2+\omega^2}$$where $\omega^2 = 2$. Note I multiplied and divided the second term by ω to get the Laplace transform to look like one in the table. The first term corresponds to $C_1 \cos \omega t$, as you noted earlier. What do you get for the second term?

 

[Rubik]

Is it right to do the second term as C₂(1/(s² + 2) which corresponds to (1/$\sqrt{2}$) sin($\sqrt{2}$t)?

 

[vela]

Sure. I just wrote it with ω on top because that's how it most likely appears in the table, rather than 1/(s²+ω²).

 

[Rubik]

Thank you so much for all your help :D I am finally beginning to understand!

 

[rude man]

Y = (s(C1))/[s2 + 2] + (C2)/[s2 + 2]

And that is as far as I can get as I am unsure what to do now?


Just use tables! Of course you have to jiggle your constants around a bit in order to accommodate the particular form your tables happen to be in.