Initial Condition (1,1)
My work as follows: First I rewrite as a first order linear
then find the integrating factor
Then multiply by integrating factor
Then product rule
And then solve for this
Am I doing something wrong here? Any help is appreciated :)
Maybe you overthought it a bit, but I think it is just this:
1 + y x-1 dy/dx
= -1 y x-2
= - y/x2 ??
I solved the de as you did and got exactly the same result.
You can check that it's correct by differentiating it and seeing that it takes you back to the original de.
\(\displaystyle y = x \ln(x) +x, \\ y' = \ln(x) + x/x + 1 =\ln(x)+2, \\ xy'=x\ln(x)+2x =( x\ln(x)+x)+x=y+x,\\ y'=y/x+1,\)
which is the original de, so with you, I think that it's correct.