Question

Find the solution of the IVP:
y''+4y=t^2+3e^t, y(0)=0, y'(0)=2.

Answer 1

can you find the characteristic equation, and the Eigen values?

Answer 2

now you need to find the particular solution of the form At^2+Bt+C+De^t

Answer 3

(at² + bt + c + de^t)'' + 4(at² + bt + c + de^t) = t² + 3e^t

Answer 4

now can you get a,b,c,d?

Answer 5

this is like partial decomp...

Answer 6

@Cacastro ? do you see what I am doing? we solved the homogenous version first, now we make up for the forcing terms

Answer 7

the general non hg solution is
c_1 cos(2t) + c_2 sin(2t) + at² + bt + c + de^t
then we use initial to find c_1 and c_2

Answer 8

(at² + bt + c + de^t)'' + 4(at² + bt + c + de^t) = t² + 3e^t
4at² + 4bt + (2a + 4c) + 5de^t = t² + 3e^t

Answer 9

I'm trying to find a,b, c and d

Answer 10

do you see that he simplified everything up to this
\[4at ^{2}+4bt+(2a+4c)+5de ^{t}=t ^{2}+3e ^{t}\]
can you look and match the coefficients to find a and b and d?

Answer 11

4a=1
4b=0
2a+4c=0
5d=3

Answer 12

a=1/4 , b=0, c=-1/8, d=1

Answer 13

think you need to check d

Answer 14

5d=3

Answer 15

d=3/5

Answer 16

good, now plug those values into the formula above and you'll have the particular solution

Answer 17

not yet...you need to solve for your other coefficiants.

Answer 18

coefficients

Answer 19

the particular solution is not the final answer

Answer 20

the solution geral is \[y(t)= C _{1} \cos 2t + C _{2} sen 2t + \frac{ 1 }{ 4 } t^2- \frac{ 1 }{ 8 } + \frac{ 3 }{ 5 }e^t\]

Answer 21

and now i have to replace for the initial conditions

Answer 22

yes?

Answer 23

yes :)

Answer 24

I found C1= -19/40 and c2= 7/310 A solução do PVI é \[- \frac{ 19 }{40 } \cos 2t + \frac{ 7 }{ 10 } \sin 2t + \frac{ 1 }{ 4 }t^2 - \frac{ 1 }{ 8 } + \frac{ 3 }{ 5 } e^t \]

Answer 25

guess i better do it to check your work, give me a few mins ok?

Answer 26

ok

Answer 27

I got something different

Answer 28

oh wait i forgot the 1/8

Answer 29

that looks right nice work

Answer 30

hmm I got 28/40 on sin

Answer 31

nm lol

Answer 32

sorry

Answer 33

you got what I got, nice work...these are long:)

Answer 34

yeah these can get really really messy;

Answer 35

Thank very much