What went wrong with my solution for a damped mass spring system?

[FHamster]

So the equation is x'' + 10x' + 64x = 0
x(0) = 1
x'(0) = 0
I get general solution of e^(-5t)(c1*cos(6.245t) + c2sin(6.245t) )
From there I get e^(-5t)cos(6.245t)+5e^(-5t)sin(6.245t)
but it wrong. What the gerbils am I doing wrong?

Thanks

 

[tiny-tim]

Welcome to PF!

Hi FHamster! Welcome to PF!

(try using the X² button just above the Reply box )

So the equation is x'' + 10x' + 64x = 0
x(0) = 1
x'(0) = 0
…
From there I get e^(-5t)cos(6.245t)+5e^(-5t)sin(6.245t)
but it wrong. What the gerbils am I doing wrong?

(do hamsters not like gerbils? )

erm … haven't you left out a 6.245 ?

 

[HallsofIvy]

So the equation is x'' + 10x' + 64x = 0
x(0) = 1
x'(0) = 0
I get general solution of e^(-5t)(c1*cos(6.245t) + c2sin(6.245t) )
From there I get e^(-5t)cos(6.245t)+5e^(-5t)sin(6.245t)
but it wrong. What the gerbils am I doing wrong?

Thanks

If $y= e^{5t}(c_1cos(6.245t)+ c_2sin(6.245t))$
then $y(0)= e^0 (c_1 cos(0)+ c_2 sin(0))= c_1= 1$ so you have that coefficient right.

$$y'= 5e^{5t}(c_1 cos(6.245t)+ c_2 sin(6.245t))+ e^{5t}(-6.245 sin(6.245 t)+6.245cos(6.245t))$$

$$y'(0)= 5e^0(c_1 cos(0)+ c_2 sin(0))+ e^0(-6.245c_1 sin(0)+ 6.245c_2 cos(0))$$
$$y'(0)= 5(c_1)+ (1)(6.245c_2)= 11.245$$
Knowing that $c_1= 1$, solve for $c_2$.

 

[FHamster]

Thanks, following some of these guidelines and doing some recalcumacations. I managed to get it right

Hi FHamster! Welcome to PF!

(try using the X² button just above the Reply box )


(do hamsters not like gerbils? )

erm … haven't you left out a 6.245 ?

Hamsters are the masterrace.

 

[blue_raver22]

for sharing your work and question! It seems like you are on the right track with finding the general solution for this damped mass spring system. However, there may be a mistake in your final solution. The general solution you provided is correct, but the specific solution that satisfies the initial conditions of x(0)=1 and x'(0)=0 is actually e^(-5t)cos(6.245t). This can be found by plugging in the initial conditions and solving for the constants c1 and c2. So the final solution should be e^(-5t)cos(6.245t). It is possible that you may have made a mistake when solving for the constants or when plugging in the initial conditions. I would recommend double checking your work and calculations to see if you can identify where the mistake may have been made. Keep up the good work!