Question

given that :
f''(x) = x + (f'(x) /x)
f'(-1)=f(1)= -1
then solve f(x)..i mean solve the differential eqn,

Answer 1

@Aryang

Answer 2

ok then it will be xf"(x)=x^2+f'(x)

Answer 3

yes..then ..??

Answer 4

leme try..
xf'(x) - f(x) = x^3 /3 + f(x) + C..is that right ?

Answer 5

now if u solve the equation by integration u will get answer

Answer 6

ya right@ar

Answer 7

but what'd be my next step ? if i integrate now, i'll have a integral(f(x)),,wont that just get stuck ?

Answer 8

now u have to substitute the value f'(-1)=f(1)= -1

Answer 9

alright,,that'll get me the value of C,,but main aim is to solve for the eqn of the curve..??

Answer 10

??

Answer 11

try this
\[ f'(x) = u \\
f''(x) = u'\]

Answer 12

there isn't f(x) ... so you can substitute f'(x) =u, this reduces to first order equation. try to solve it step wise.

Answer 13

then what do i substitute for x ?

Answer 14

you don't need to substitute it for x. 'u' will be function of x.

Answer 15

leme try..

x u' = x^2 + u
then integrate..right?

Answer 16

yep ... this is linear DE. find u
\[ u = f'(x) = .... \]

Answer 17

okay,.
u - f(x) = (x^3 /3) + f(x)
then?? wont it be stuck afterwards??

Answer 18

you mean the integration factor thing ?

Answer 19

\[ \large u = c e^{\int {1 \over x }dx} dx + e^{\int {1 \over x }dx} \int e^{-\int {1 \over x }dx} {1 \over x} dx \]

Answer 20

how'd you get that ?

Answer 21

http://www.sosmath.com/diffeq/first/lineareq/lineareq.html

Answer 22

integration factor was 1/x right?
you arranged it as
u' - u/x = x ?

Answer 23

yep

Answer 24

so simplified form is
u/x = x + C or
u = x^2 + Cx
?

Answer 25

or
u = x^2 + 2x ??

Answer 26

ahh..,i get it,,integrating at the next step directly solves for f(x)..thanks @experimentX

Answer 27

yep
\[ g\left (x, f^{n-1}x , f^{n}x \right) = 0\]
solve this type of equation using this technique.

Answer 28

i get it,,i know all of these,,just didnt strike me then,,thanks again..

Answer 29

leme try an alternative method..
d/dx ( f'(x) /x) = (x f''(x) - f'(x) )/x^2
you are given
f''(x) = x + (f'(x) /x)
or x f''(x) - f'(x) = x^2

thus on making this simple substitution,
we have
d/dx(f'(x) /x) = 1
or
f'(x) = x^2 +Cx

ofcorse, you can do further.. :-)

Answer 30

aha,,nice..thanks!! :)