So I have two problems that ask me to separate variables:
1: du/dx * x(x+1) = u^2 u(1)=1
2: dy/dt=y(2-y) Initial condition y(0)=1
The Attempt at a Solution
For #1 i get each variable on one side and have integral(u^-2du)=integral(dx/x(x+1)
I use partial fractions to break up the x side, coefficient are 1 and -1. Now I have:
-1/u=ln(x/x+1) +C ==> u=-1/(ln(x)-ln(x+1)) + C. And evaluated at (1,1) C=ln2.
that would give us, u=-1/(ln(x)-ln(x+1)) + ln2, however the answer in my book gives u=-1/((ln2x/x+1)-1) I'm having trouble seeing where the minus one in the denominator comes from can someone show me an explanation?
For #2 i get integral(dy/y(2-y)) = integral(dt) I again use partial fractions to break up the y side, giving coefficients of 1/2. I'm left with 1/2(ln(y) + ln(2-y))=t. I'm having trouble isolating y on one side. I need a point in the right direction.
Thanks for the help!