#University Exam Question Differential Equation
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river wind
okay, this might be a new method for you, but this was helpful for me when i started out differential equations
we have that $y'=\frac{x+y+1}{2x+y+1}$
bold gardenBOT
dark matter
river wind
now we apply the property of homogeny in that $f(cx,cy)=c^nf(x,y)$ then the equation is a degree $n$ homogenous equation
bold gardenBOT
dark matter
river wind
so we test and get $y'=\frac{cx+cy+1}{2cx+cy+1}$
bold gardenBOT
dark matter
river wind
**dark matter**
(remember that $n\in\bN$ here)
bold gardenBOT
dark matter
river wind
**dark matter**
but this will never result in the equation replicating itself in some c^n form so the equation is actually not homogenous
river wind
(ok mb @rotund trout its homogenous with z=y+1 i forgot to say this)
(i solved away from device)
yeah
yes
so z'=y'
thus $z'=\frac{x+z}{2x+z}$
bold gardenBOT
dark matter
river wind
here we can just divide by x on the RHS numerator and denominator and use standard subsitutions
yeah
lol
so jst use another variable to let w=z/x
z=wx
z'=w+w'x
and that should do it
you should get an equation in some form $w'=\frac{1}{x}f(w)$ but i am probably mistaken
bold gardenBOT
dark matter
river wind
**dark matter**
this is obviously seperable
small lodgeBOT
@rotund trout