What is a solution to the differential equation $dy/dx=x/(x+y)$ ?

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Answer 1 · 2017-01-31T12:04:40

See below.

Making $y = lambda x$ and knowing that

$dy=lambda dx + x d lambda->(dy)/(dx)=lambda + x(d lambda)/(dx)$

we get at

$lambda + x (d lambda)/(dx)=1/(lambda+1)$ or

$x (d lambda)/(dx)=-(lambda^2+lambda-1)/(lambda+1)$ and now this differential equation is separable

$(dx)/x=-((lambda+1)d lambda)/(lambda^2+lambda-1)$

so

$(dx)/x = -((lambda+1)d lambda)/((lambda+1/2)^2-5/4)$

giving

$logx=-1/10((sqrt(5)+5)log(sqrt5-1-2lambda)-(sqrt(5)-5)log(sqrt5+1+2lambda))+C$

Now substituting,

$logx=-1/10((sqrt(5)+5)log(sqrt5-1-2y/x)-(sqrt(5)-5)log(sqrt5+1+2y/x))+C$

so the solution appears in implicit form

$G(y(x),x,C)=0$

What is a solution to the differential equation dy/dx=x/(x+y)dy/dx=x/(x+y)?