Question

given the differential dy/dx=(x+y+1)/(x+y+3).
by using the substitution z=x+y+1,find the solution of the equation

Answer 1

Replace x+y by z-1 up and down
Then continue :)

Answer 2

what is dz/dx?

Answer 3

1+(dy/dx)

Answer 4

btw this question under topic of ODE of first order

Answer 5

so dy/dx = dz/dx - 1

Answer 6

What is the equation now? (in terms on dz/dx, and z)

Answer 7

\[\frac{dy}{dx}=\frac{x+y+1}{x+y+3}\]
letting \(z=x+y+1\);
\[\dfrac{dz}{dx}=1+\dfrac{dy}{dx}\\\dfrac{dy}{dx}=\dfrac{dz}{dx}-1\]

Substituting these into the DE
\[\dfrac{dz}{dx}-1=\frac{z}{z+2}\]