Question

Solve the differential Equation
\[\frac{dy}{dx}+\frac{2x−y−4}{ 2x − y + 5}=0\]

Answer 1

Try writing v = 2x - y. That may simplify by a bunch.

Answer 2

Then: dv/dx = 2 - dy/dx. Rewrite dy/dx as 2 - dv/dx :-)

Answer 3

That becomes a separable DE.

Answer 4

\[2-\frac{\text d v}{\text d x}+\frac{v-4}{v+5}=0\]
this looks seperable

Answer 5

yup :-)

Answer 6

thankyou bmp

Answer 7

No problem :-)

Answer 8

\[\frac{\text d v}{\text d x}=2+\frac {v-4}{v+5}\]\[\frac{\text d v}{\text d x}=\frac{2(v+5)+(v-4)}{v+5}\]\[\frac{\text d v}{\text d x}=\frac {3v+6}{v+5}\]\[\frac{\text d v}{\text d x}=\frac {3(v+2)}{v+5}\]\[\frac{\text d x}{\text d v}=\frac {v+5}{3(v+2)}\]\[\text d x=\frac {v+5}{3(v+2)}\text d v\]\[\int\text d x=\frac 1 3\int\frac {v+5}{v+2}\text d v\]

right so far?

Answer 9

\[x=\frac13 \int\frac {v}{v+2}+\frac {5}{v+2}\text d v\]

Answer 10

let x = X+h and y = Y+k <--- reduce it into homogeneous differential equation.

Answer 11

@experimentX
are you suggesting that the method i have working for, is not the bet way to answer this question?

Answer 12

I haven't looked ... i remembered that i solved these kinds of differential equation by reducing it to homogeneous differential equation.
2h - k - 4 =0
2h - k + 5 =0
Oo ... looks like your method is right!!!

Answer 13

it could be that both methods work but i might try the first method to the end first

Answer 14

\[x=\frac13 \int\frac {v}{v+2}+\frac {5}{v+2}\text d v\]\[x=\frac{1}{3} \left( v-2\log(v+2)+5\log(v+2)\right) + c_1\]

Answer 15

\[x=\frac{1}{3} \left( v+3\log(v+2)\right) + c_1\]\[x= \frac{(2x-y)}{3}+\log((2x-y)+2) + c_1\]\[x= \frac{(2x-y)}{3}+\log(2x-y+2) + c_1\]

Answer 16

now what should i do?

Answer 17

\[x= \frac{(2x-y)}{3}+\log(2x-y+2) + c_1\]

this is the solution.. you cannot do anything beyond this..

Answer 18

ok cool