[itex]\displaystyle e^{-1/x-\ln(x)+C}=e^{-1/x}e^{-\ln|x|}e^C[/itex]bdh2991 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?
Separation of variables is a method used to solve differential equations by separating the variables into two or more equations with only one variable in each equation.
Separation of variables is important because it allows us to solve complex differential equations that cannot be solved using traditional methods. It is a powerful tool in many areas of science and engineering.
The steps involved in separation of variables include identifying the variables in the differential equation, separating the variables into two or more equations, solving each equation separately, and then combining the solutions to obtain the final solution.
Separation of variables can help solve a wide range of problems, including problems in physics, engineering, economics, and biology. It is particularly useful in solving problems involving heat transfer, diffusion, and wave propagation.
Yes, there are some limitations to separation of variables. It can only be used to solve linear partial differential equations with separable variables. It is also not always possible to find a solution using this method, and in some cases, the solution may not be unique.