The discussion focuses on solving the differential equation x²(dy/dx) = y - xy using the method of separation of variables, with the initial condition y(-1) = -1. The user initially derived the solution y = e^(-1/x - ln x + C) but struggled to match it with the book's answer, y = e^(-(1 + 1/x))/x. The key insight provided was the algebraic manipulation of the exponential function, specifically recognizing that e^(-ln x) simplifies to 1/x, leading to the correct form of the solution.
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\displaystyle e^{-1/x-\ln(x)+C}=e^{-1/x}e^{-\ln|x|}e^Cbdh2991 said:Homework Statement
use separation of variables to solve the differential equation x^2dy/dx=y-xy
with the initial condition of y(-1)=-1
The Attempt at a Solution
after i separated and integrated i got the answer y=e^(-1/x-lnx+c)
the answer in the book is y=e^-(1+1/x)/x
i can't figure out how they got to that even after i plugged in -1 for x and y
some help would be greatly appreciated
bdh2991 said:well i think it would be x^-1 so then that would give me the x in the denominator then plugging in the initial values i should get c=1?