## S_Student 2 years ago Can anyone solve it Laplace L(3t^2+3t^3+e^t+sin3t)

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1. UnkleRhaukus

$\mathcal L\{3t^2+3t^3+e^t+\sin3t\}$$\qquad=3\mathcal L\{t^2\}+3\mathcal L\{t^3\}+\mathcal L\{e^t\}+\mathcal L\{\sin 3t\}$

2. UnkleRhaukus

$\boxed{\mathcal L\big\{t^n\big\}=\dfrac{\Gamma(n+1)}{s^{n+1}}}\qquad\boxed{\mathcal L\big\{e^{-at}\big\}=\dfrac{1}{s+a}}\qquad\boxed{\mathcal L \big\{\sin(bt) \big\}=\dfrac{b}{s^2+b^2}}$

3. S_Student

thank you so much dear..

4. UnkleRhaukus

$\color{teal} {\ddot\smile}$

5. S_Student

can i ask one more question..?

6. UnkleRhaukus

yes

7. S_Student

ok thnx.. d^3y/dt^3 + d^2y/dt^2 + dy/dt = sint

8. UnkleRhaukus

the laplace of?

9. S_Student

yup..

10. UnkleRhaukus

use $\boxed{\mathcal L\big\{f'(t)\big\}=sF(s)-f(0)}$$\boxed{\mathcal L\big\{f^n(t)\big\}=s^nF(s)-s^{n-1}f(0)-s^{n-2}f'(0)\dots-sf^{n-2}(0)-f^{n-1}(0)}$

11. hartnn

since initial conditions are not given, assume them to be 0.

12. UnkleRhaukus

hmm, the question should state the initial conditions

13. hartnn

which gives you, $$\boxed{\mathcal L\big\{f^n(t)\big\}=s^nF(s)}$$

14. hartnn

when not given, we can safely assume then to be 0.

15. UnkleRhaukus

i wouldn't assume that, i would have initial conditions in my final result

16. abb0t

hmmm....verrrry interesting symbols ya'll got there.

17. S_Student

i dont understand :(

18. Tushara

do u have the initial conditions?

19. S_Student

yup

20. Tushara

21. S_Student

Laplace d^3y/dt^3 + d^2y/dt^2 + dy/dt = sint

22. Tushara

there shud be more... r u give what y(0) is?

23. hartnn

dy/dt = f'(t) d^2y/dt^2 = f'' (t) d^3y/dt^3 = f'''(t) and then use, $$\boxed{\mathcal L\{f^n(t)\big\}=s^nF(s)-s^{n-1}f(0)-s^{n-2}f'(0)\dots-sf^{n-2}(0)-f^{n-1}(0)}$$

24. Tushara

given*

25. S_Student

DOnt know y(0) :(

26. S_Student

oh thnx hartnn

27. S_Student

i want thats,,step by step so thnx

28. S_Student

thanks to all who try to help me..

29. Tushara

i hope u can solve it now.... this might help, check out example 3: http://www2.fiu.edu/~aladrog/LaplaceTransDifferentialEq.pdf

30. S_Student

OK...thnx :)