Question

I'm not sure how to solve this equation

y' (2x - 2y - 4) = 2x + 7y + 5

Answer 1

we are studying linear differential equations at the moment

Answer 2

Umm what methods do you know as of yet

Answer 3

Are you familiar with the method Exact equations?

Answer 4

we've been doing integrating factors and homogeneous equations, this question was the
last one the set homework.

Answer 5

exact equations? no

Answer 6

How about substitutions?

Answer 7

ah - it was mentioned in our last lecture - do you think it would help here?

Answer 8

Ya I do think so actually

Answer 9

Wait let me see one sec

Answer 10

i'll havev to check in my written notes.

Answer 11

i must admit i'm struggling a bit with this topic

Answer 12

You can definitely use substitution just gotta figure it out

Answer 13

I am having a hard time finding a substitution

Answer 14

you can write this equation as
(2x - 2y - 4) dy = (2x + 7y + 5) dx

have you seen the technique where you shift the origin so that the coefficients are zero:

solve
2x - 2y - 4 =0
2x + 7y =0
to find (1,-1)

let u= x -1 and v = y+1
to get
(2u - 2v) dv = (2u + 7v) du

this can be solved using another substitution:
v = w u , dv = w du + u dw

Answer 15

another method is use the substitution
u = 2x -2y -4 du = 2 dx -2 dy
v = 2x + 7y + 5 and dv = 2 dx + 7 dy
solve for dx and dy in terms of du and dv

this should get you to something you can solve