## anonymous 5 years ago y" + 3y' + 2y = 6 y(0)=0 y'(0) = 2 with laplace

1. anonymous

y'' = sy^2 - sy(0) -y'(0)

2. anonymous

y' = sy - y(0)

3. anonymous

(sy^2 -2) +3(sy) +2y = 6

4. anonymous

5. anonymous

yes, im stuck in integral (6+2s) / s(s+2)(s+1)

6. anonymous

First write everything in terms of the laplace transform. THen solve the equation by converting back.

7. anonymous

no, so you got y to be that

8. anonymous

Use particla fractions.

9. anonymous

then you must use partial fractions

10. anonymous

6+2s = As + B(s+2)+C(s+1) then?

11. anonymous

= 3/s -4/(s+1) +1/(s+2)

12. anonymous

I created and went straight to wolframa for the partial fractions

13. anonymous

cheated

14. anonymous

partial fractions are very standard

15. anonymous

then you convert them back

16. anonymous

the 3/s goes to 3 from memory

17. anonymous

and the other two are time shifted exponentials

18. anonymous

Laplace is all about matching and partial fractions, at least in solving simple ODE systems.

19. anonymous

6+2s = As + B(s+2)+C(s+1) s=-1 --> 4 = -A+B s=-2 --> 2 = -2A-C for the last s what number should i choose?

20. anonymous

1/(s-a) = e^(at) ( I googled this lol )

21. anonymous

s=0 lol

22. anonymous

remember you can pick any value for s, just that some values will make the simultaneous eqns alot easierto solve

23. anonymous

6+2s = As + B(s+2)+C(s+1) s=-1 --> 4 = -A+B s=-2 --> 2 = -2A-C s=0 --> 6 = 2B+C ---------------------- 4+A = B 6 = 2(4+A)+C 6 = 8 +2A+C elimination -2 = 2A+C 2 = -2A -C infinity?

24. anonymous

y= 3-4e^(-t) + e^(-2t)

25. anonymous

you make no sense at all lol

26. anonymous

this is why people need to pay attention in high school and first year uni maths course , so they absolutely hammer in the basics

27. anonymous

could u help me?

28. anonymous

its just simultaneous eqns , takes for ever, you need to set up a matrix etc.

29. anonymous

it will take me like 10mins to type it up , I aint doing it lol

30. anonymous

matrix? really?

31. anonymous

I can see that you reach the point $$Y(s)=\frac{2s+6}{s(s+1)(s+2)}$$. Now we should use partial fractions to write the expression in a form that can be easily to find its inverse laplace transform. That's ${2s+6 \over s(s+1)(s+2)}={a \over s}+{b \over s+1}+{c \over s+2}$. Multiplying both sides by $$s(s+1)(s+2)$$ gives: $2s+6=a(s+1)(s+2)+bs(s+2)+cs(s+1)$ Plugging $$s=0$$ gives $$2a=6 \implies a=3$$; $$s=-1$$ gives $$-b=4 \implies b=-4$$, and $$s=-2$$ gives $$2c=2 \implies c=1$$. So, $$Y(s)=3\frac{1}{s}-\frac{4}{s+1}+\frac{1}{s+2}$$. Hence $$y(t)=3-4e^{-t}+e^{-2t}$$.

32. anonymous

Hello suzi!! Does the answer make sense to you?

33. anonymous

thank you anwara

34. anonymous

You're welcome!