Question

Solve the ODE
\[\frac{\text d y}{\text dx}=-\frac{2x+3y+1}{4x+6y+1}\]

Answer 1

\[\text{let }v=2x+3y\]\[\frac{\text d v}{\text d x}=2+3\frac{\text d y}{\text dx}\]\[\frac{\text d y}{\text dx}=\frac13\left(2-\frac{\text d v}{\text d x}\right)\]\[\frac13\left(2-\frac{\text d v}{\text d x}\right)=-\frac{v+2}{2v+1}\]\[\frac{\text d v}{\text d x}=2+\frac{3(v+2)}{2v+1}\]\[\frac{\text d v}{\text d x}=\frac{2(2v+1)+3(v+2)}{2v+1}\]\[\frac{\text d v}{\text d x}=\frac{7v+8}{2v+1}\]

Answer 2

From
\[\frac{\text d v}{\text d x}=2+3\frac{\text d y}{\text dx}\]
wouldn't dy/dx is

\[\frac{dy}{dx}=\frac{1}{3}(\frac{dv}{dx}-2)\]

Answer 3

yeah totally,
thanks

Answer 4

yw :)

Answer 5

\[\frac{\text d y}{\text dx}=-\frac{2x+3y+1}{4x+6y+1}\]\[\text{let }v=2x+3y\]\[\frac{\text d v}{\text d x}=2+3\frac{\text d y}{\text dx}\]\[\frac{\text d y}{\text dx}=\frac13\left(\frac{\text d v}{\text d x}-2\right)\]\[\frac13\left(\frac{\text d v}{\text d x}-2\right)=-\frac{v+1}{2v+1}\]\[\frac{\text d v}{\text d x}=\frac{2(2v+1)-3(v+1)}{2v+1}\]\[\frac{\text d v}{\text d x}=\frac{v-1}{2v+1}\]

Answer 6

any mistakes this time?

Answer 7

Good to go :)

Answer 8

ill work on this again tomorrow, no more for today

Answer 9

im am pretty sure i have solved it now

Answer 10

There's a bracket mistake in line 8

Answer 11

oh yes,
thank you,
i think ill take that odd bracket out