Question

Differential equation

Answer 1

\[y^{(4)}-y=0\] can be written as \[y=c_1 cost+c_2sint+c_3 cosht+c_4sinht\]

Answer 2

a fourth order differential equation !!

Answer 3

Yup

Answer 4

I'm not even sure how to begin with this haha, the characteristic equation \[R^4-1 = 0\]

Answer 5

This means nothinggg

Answer 6

I don't think that I've ever seen anyone solve a 4th order DE
Of course that won't work :)

Answer 7

\[(r-1)(r+1)(r^2+1)=0\]

Answer 8

Nopeee

Answer 9

Oh wait.. so the roots are +1,-1,-i,+i

Answer 10

everything that works for second order must work for higher orders too

Answer 11

Well we could go backwards.
Find y'''' using the expression given and then substitute to see if it equals y ?

Answer 12

I didn't know that you could use the characteristic equation for 3rd and 4th order DE's

Answer 13

\[y=c_1e^{-t}+c_2e^{t}+c_3\cos(t)+c_4\sin(t)\]

Answer 14

wait can we use euler's eqn

Answer 15

formula*

Answer 16

yes and then we get your original expression with cosh and sinh !!!

Answer 17

Ahah I wasn't sure about the hyperbolics

Answer 18

yeah, it works! well done, easy after all because the characteristic eqn can be used

Answer 19

\[e^x = coshx+sinhx\] just googled

Answer 20

I think that definitely works then

Answer 21

it would be easier if you use knowledge about linearity of solutions

Answer 22

since \(e^t\) and \(e^{-t}\) are two independent solutions of the equation \(y^{(4)}-y=0\),
would you agree that "every" linear combination of them is also a solution ?

Answer 23

That would make sense

Answer 24

would you also agree that all below satisfy the DE ?

\(c_1e^{t} \)

\(c_1e^{-t} \)

\(c_2e^{t} \)

\(c_2e^{-t} \)

Answer 25

therefore their sum (linear combination) also satisfies the DE :
\[c_1e^t + c_1e^{-t} + c_2e^t + c_2e^{-t}\]

Answer 26

rearrange and get
\(c_1(e^t + e^{-t}) + c_2(e^t + e^{-t})\)

=\(C_1\cosh t+C_2 \sinh t \)

Answer 27

just some algebra rearrangement..

Answer 28

I would've never thought of this, nice one

Answer 29

Using your method, what about the C1cost and C2sint terms ?

Answer 30

they just carry along..

Answer 31

You can do similar

Answer 32

u get cos and sin from the imaginary roots

Answer 33

i like these wolfram solutions because it provides nice reasons for each and every step

Answer 34

what do you mean they "just carry along"?

Answer 35

`cost` and `sint` are other two "independent" solutions that satisfy the given DE

Answer 36

since the differential equation is of order 4, we need to find exactly 4 independent solutions to write out the general solution

Answer 37

OK!! The wolfram thing is really good. Explains everything just nicely and what an elegant solution.

Answer 38

So this is how you answer the questions :)

Answer 39

Do you have to pay for that feature? i.e. the solution of DE's

Answer 40

Ok that just means, I got lucky by using Euler's method directly haha

Answer 41

formula

Answer 42

No you didn't get lucky! It was a very intelligent guess which led to the right answer. With ganeshies help regarding the characteristic eqn validity

Answer 43

@alekos you may create an wolfram account here
http://www.wolframalpha.com/pro/trial.signin.html

account will have all the pro features (like solution steps, downloading images etc) for 7 days. no credit card necessary to try..

Answer 44

How much for the pro version?

Answer 45

I mean we were sort of forced to solve this characteristic eqn from what I'm getting, since \[e^{\lambda x} \neq 0\]

Answer 46

less than $100 per year
http://www.wolframalpha.com/pro/

Answer 47

around $60 i think..

Answer 48

Everyone should pitch in $5 and get this sucker

Answer 49

it cannot provide solution for every problem though..
if the problem is moderately tough, it gives up cheaply..

Answer 50

oh, I see. Nevertheless $60 is not bad

Answer 51

Too bad it's not a one time thing, I would totally buy it

Answer 52

I'd be down xD

Answer 53

even $10

Answer 54

I am good with creating a new account and use trial period once in a while..

Answer 55

Hahaha

Answer 56

I haven't legit used it for a while other than letting it factor for me

Answer 57

@alekos =D just make several random accounts there until u become rich then estimate how much u spent over there, send them some donation and say sorry xD

Answer 58

So how many times can you create an account? or do you use different email addresses each time?

Answer 59

you may use below site for on the fly email id creation
http://www.fakemailgenerator.com/

Answer 60

If they ever get like a one time payment thing I'll do it, but not monthly kind of thing..haha ganeshie I knew you had a generator or something

Answer 61

I could not imagine you making random accounts every time

Answer 62

you will need to provide a different email id everytime you create a new account

Answer 63

I had no idea that website existed! Thanks ganeshie

Answer 64

Woah really...the joy you must have right now

Answer 65

You can do lots of cool stuff with it for example http://www.wolframalpha.com/input/?i=Albert+Einstein%2C+Paul+Dirac%2C+Richard+Feynman&lk=3

Answer 66

It's not just limited to math :P

Answer 67

http://www.wolframalpha.com/input/?i=Scarlett+Johansson

Answer 68

X'D

Answer 69

And there I was thinking that it could only solve maths problems. My god!

Answer 70

By the way ganeshie, the fake mail generator works a treat! thanks!

Answer 71

http://www.wolframalpha.com/input/?i=Randeep+Hooda

Answer 72

congrats ! have fun :)

Answer 73

wondering which data base it use for people like actors/politics ... ect
like any one could modify(as wiki ) ?

Answer 74

this one is weird though


http://www.wolframalpha.com/input/?i=ikram+ibrahim

Answer 75

this is a dangerous feature
http://www.wolframalpha.com/input/?i=where+am+i

Answer 76

interesting
http://prntscr.com/90e2by

Answer 77

it use IP address but seems can't match my place though

Answer 78

hax

Answer 79

not in particulate should't it use some place around instead of 100km away ?

Answer 80

ganeshie where is the 7 day trial I can't seem to find it, used to be on accounts

Answer 81

there are 5 people around the world been called ikram per year =D // according to wolfram, it make me feels special though

Answer 82

that was removed i guess...
go here to create a pro account directly
http://www.wolframalpha.com/pro/trial.signin.html

Answer 83

Ah thanks, yeah I could see why. People probably abused it..

Answer 84

*cough*

Answer 85

hmm it could be seen as an abuse, but the pro feature is completely useless to me as i don't take math seriously and im done with my studies long ago..

Answer 86

You not taking math seriously?! :P

Answer 87

wolfram is math project they didn't need money ( i always wondered why they ask for pro accounts) still a mystery they even don't take much money also they don't solve more questions that the one u see when u are regular user.

Answer 88

Yeah I personally just use it as a calculator to check

Answer 89

it's like nothing interesting with pro accounts over there, as they only use known algorithms/data base, it's not like they would give u extra executing time for interesting questions or even solve high mathematics ( pure direction ).

Answer 90

http://www.wolframalpha.com/input/?i=why+did+the+chicken+cross+the+mobius+strip&lk=3

Answer 91

i like that sense og humor =)

Answer 92

Hopefully google just buys wolframalpha and everyone gets it for free

Answer 93

no no no NO HELL NO xD

Answer 94

i wanna buy it first :P

Answer 95

Well you better hope you're not a math major then! xD

Answer 96

not at all, no regret =)

Answer 97

2018 they will buy it for $6.3 billion