Question

find y for y'=-2(x-2y+1)/(5x-y-4)

Answer 1

y is y(x)

Answer 2

this is a double integral problem

Answer 3

or do you want to integrate only with respect to x

Answer 4

y is a function of x

Answer 5

No, you can't just do the second integral with respect to x and treat the y as constants, precisely because y is also a function of x.

Answer 6

Yes. This looks like it wants to be a homogeneous equation in the sense that the variable we're interested in using is something like this
v = (y+c)/x

This is a nice substitution, because it implied xv = y + c and thus differentiating
y' = xv' + v

The trick is to rearrange
-2(x-2y+1)/(5x-y-4)
somehow to use that substitution

Answer 7

i am back

Answer 8

i apologize for the silly mistake i made

Answer 9

so it seems we are dealing with a differential equation here

Answer 10

i was thinking in terms of double integral

Answer 11

the answer for the differential equation would be

Answer 12

\[f(x) = 1-\frac{ 1 }{ 2 }\frac{ 2C(x-1)+1+\sqrt{12C(x-1)} }{ C }\]

Answer 13

where C is a constant

Answer 14

i will delete the mistakes i made so no one waste time reading them

Answer 15

no problem , everybody makes mistakes, but how do I get that result ? Also I can't find a way to rearange the -2(x-2y+1)/(5x-y-4) equation for the substitution to be useful

Answer 16

do you agree with the answer first so i show you the steps

Answer 17

I don't know the answer and I don't know how you got that answer :) the steps would be really useful thanks

Answer 18

sure will do

Answer 19

it will take a while to type the answer and wish me good luck i don't get phone call from my students

Answer 20

ok back sorry for the long delay due to phone calls

Answer 21

step 01: let y = xv

Answer 22

02: dy = xdv + vdx

Answer 23

03: (5x-y-4)dy =(-2x+4y-2)dx

Answer 24

04: (5x-y-4)(xdv+vdx)=(-2x+4xv-2)dx

Answer 25

let me know if you are following

Answer 26

now we have to simplify the equation in terms of v and x, dv, and dx

Answer 27

let me know when you are back

Answer 28

I'm back , I'm reading the steps you wrote right now

Answer 29

\[5x ^{2}dv-x^{2}vdv-4xdv-5xvdx-xv^{2}dx-4vdx=...\]

Answer 30

all of that equals to the other side which is:

Answer 31

\[-2xdx+4xvdx-2dx\]

Answer 32

i think you can do it from here put the dx on one side and the dv on the other side

Answer 33

now there are many ways to solve this problem using partial fraction, also the sinh and cosh functions but i am not sure if you have done such problems

Answer 34

there is another substitution that you could do which is u= ax + by +k

Answer 35

thank you very much for the help

Answer 36

you are more than welcome, thanks for your kindness

Answer 37

please tell ur teacher to kindly write the function f(x,y) so i don't make a mistake

Answer 38

hehehe