I am trying to solve a homogeneous, first-order, linear, ordinary differential equation but am running into what I am sure is the wrong answer. However I can't identify what is wrong with my working?!
. Let $z=x/y$, so that $y=x/z$ and Then
Thus .
However, my book: *A Modern Introduction to Differential Equations 2nd edition by Henry Ricardo* says "this first-order equation is homogeneous and can be solved implicitly".
They pursued the method of letting $z=y/x$ and obtained the solution as
.
Why are the two different substitutions, which should both be suitable, giving me two different answers?
Let V = y/x. V + xdv/dx = -1 + V/1+V, xdv/dx = -1-V^2/1+V, -(1+V/1+V^2)dv = dx/x, -arctan(V) - 1/2ln|1+V^2| = ln|x| + C, 2arctan(y/x) + ln|1+y^2/x^2| + 2ln|x| - C = 0,
finally, 2arctan(y/x) + ln|x^2+y^2| - C = 0.