IIT JEE , AIEEE Forum - Mathematics , Integral Calculus

elastiboysai

solve
d³y/dx³+d²y/dx²-2dy/dx=0.
give me a proper soln not intuitive guesses like e^x

spideyunlimited

hmmm is it...
y + y^2 / 2 - y^3 / 3 = kx^2 + cx + d

?
k, c and d all are constants.

elastiboysai

no dude

das not the ans.

spideyunlimited

d^3y/dx^3 + d^2y/dx^2 - 2dy/dx = 0

d^3y/dx^3 + d^2y/dx^2 = 2dy/dx

[ d^3y/dx^3 + d^2y/dx^2 ] / 2 = dy/dx

now LHS is average of 2 terms...
if RHS is the average of the terms in the numerator of LHS, then it implies from it that :
Either ...
1) 1 term is larger than and 1 term is smaller than RHS term by same amount.
or 2) both terms are the same in value.

(Example : (3 + 5) / 2 = 4
or (4 + 4) / 2 = 4
)

But (1) cannot b true!!!
since derivative of an expression cannot be larger in value than the expression itself.. it is always less than equal to.

Hence we are left with (2)
so .... d^3y/dx^3 = d^2y/dx^2 = dy/dx
so the value remains the same no matter wat order derivative it is.
And only one such expression i know :
e^x
:)

elastiboysai

Spidey,

Sorry, but you are completely wrong

Who said derivative of a function<the function???????

x^4-1

derivative at x=1 > fn at x=1

so ur line of reasoning is wrong!!!

ayshwarya

hey den can u say wats d ans frnd

ayshwarya

hey elastiboysai is d ans

1/3 y^3 +xy^2 -2xy +c/2x^2 +c1x +c2 =0 is it rite if rite rate me i wud surely giv d ans

elastiboysai

No

das not the ans

spideyunlimited

EDIT..
we cannot have one derivative being greater than the expression and the next derivative of that as lesser than the expression

ayshwarya

tell me d ans my head is bursting

karthik2007

Ae^x+Bcos

2 x + Csin

2 x ?

amrishsharma007

hi buddy i m an eng. stdnt & giving u easy technique 2 solve this ttype of de

first consider dy/dx=D,d2y/dx2=D2 and so on

now brek the ans. into cf (complimentary function)& pi(particular integral)

ci can b cal. as follow

1st form auxillary eqn.like considering dy/dx=m now fctorize the eqn by equating it 2 0

like ur ques m3+m2-2m=0

m(m2+m-2)=0

m=0,1,2

now cf=(constt.1)eox+(constt2)e1x+(consttt.3)e2x

so ur ans. is y=cf

now try dis mthd 4 anthr de.

karthik2007

Sorry... made a mistake...

Ans is A+Be^x+C/e^2x

elastiboysai

Right ans Karthik

did u also do it by the method amrish sharma gave ?

karthik2007

College student da

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