Solving y'(x)=(x+y(x)+1)/(x-y(x)+2)

[PauxX]

At my Analysis class we were assigned to solve the following equation: y'(x)=(x+y(x)+1)/(x-y(x)+2).
Since we are studying Homogeneous equations I was trying to find some kind of substitution but I couldn't find any.
We solved at the seminar y'(x)=(x+y(x))/(x-y(x)). Is this somehow related to my problem?
When I solved the 2nd equation I denoted z=y/x and y'=z+x*z'.
After that I replaced in the equation.
I got x*z'+z=(1+z)/(1-z)
=>z' *((1-z)/1+z^2))=1/x.
After that I integrated and replaced in the result z with y/x and find y.
I tried to apply the same method to my problem but of course it doesn't really work since I have 1 and 2 there.
Do you have any ideas?

 

[Deleted member 4993]

At my Analysis class we were assigned to solve the following equation: y'(x)=(x+y(x)+1)/(x-y(x)+2).
Since we are studying Homogeneous equations I was trying to find some kind of substitution but I couldn't find any.
We solved at the seminar y'(x)=(x+y(x))/(x-y(x)). Is this somehow related to my problem?
When I solved the 2nd equation I denoted z=y/x and y'=z+x*z'.
After that I replaced in the equation.
I got x*z'+z=(1+z)/(1-z)
=>z' *((1-z)/1+z^2))=1/x.
After that I integrated and replaced in the result z with y/x and find y.
I tried to apply the same method to my problem but of course it doesn't really work since I have 1 and 2 there.
Do you have any ideas?

Do a substitution: v(x) = x + 1.5 and u(x) = y(x) - 0.5. Then dy/dx = du/dv * dy/du * dv/dx → du/dv = (u+v)/(v-u) .................. Now solve for u(v) and from that y(x).

 

[Ishuda]

At my Analysis class we were assigned to solve the following equation: y'(x)=(x+y(x)+1)/(x-y(x)+2).
Since we are studying Homogeneous equations I was trying to find some kind of substitution but I couldn't find any.
We solved at the seminar y'(x)=(x+y(x))/(x-y(x)). Is this somehow related to my problem?
When I solved the 2nd equation I denoted z=y/x and y'=z+x*z'.
After that I replaced in the equation.
I got x*z'+z=(1+z)/(1-z)
=>z' *((1-z)/1+z^2))=1/x.
After that I integrated and replaced in the result z with y/x and find y.
I tried to apply the same method to my problem but of course it doesn't really work since I have 1 and 2 there.
Do you have any ideas?

Another method similar to the solution studied at the seminar is to let
y = a + (x+b) z
or, if you wish,
z = (y+A) / (x+B)
and start working through the equations to see if you can find a 'nice' pair for a and b. Hint: They will resemble the constants given by Subhotosh Khan in his definitions of u and v.

 

[PauxX]

Thank you very much for your help,Subhotosh Khan and Ishuda!!After seeing your ideas I realised it wasn't as complicated as I thought. Anyway thank you guys!You are great!!!