# Thread: Simple Separation of Variable Problem

1. ## Simple Separation of Variable Problem

I'm just starting a course in DiffEQ and we have a simple separation of variables problems.

The problem is to change the dy/dt equation into a simple y(t) equation (example dy/dt=ty becomes y=k*e^(t^2)) where k is any real number).

The problem I am stuck on is dy/dt=t/(y+y*t^2). Here is my work.
dy/dt=t/(y(1+t^2))
y*dy=[t/(1+t^2)]*dt
Integrate both sides
1/2y^2=1/2[ln|t^2+1|]+C where C is a constant
y^2=ln|t^2+1| + C where C is a constant
y = sqrt (ln|t^2+1| + C) where C is a constant and the sqrt is either positive or negative.

The book gives an answer of y(t)=sqrt(ln|k(t^2+1)|) where k is any real number and the sqrt is either positive or negative.

I understand most of the steps, but why do I get ln|t^2+1| + C inside the square root while the book gets ln|k(t^2+1)| inside the square root?

2. Originally Posted by David_is_a_LOSTaway
I'm just starting a course in DiffEQ and we have a simple separation of variables problems.

The problem is to change the dy/dt equation into a simple y(t) equation (example dy/dt=ty becomes y=k*e^(t^2)) where k is any real number).

The problem I am stuck on is dy/dt=t/(y+y*t^2). Here is my work.
dy/dt=t/(y(1+t^2))
y*dy=[t/(1+t^2)]*dt
Integrate both sides
1/2y^2=1/2[ln|t^2+1|]+C where C is a constant
y^2=ln|t^2+1| + C where C is a constant
y = sqrt (ln|t^2+1| + C) where C is a constant and the sqrt is either positive or negative.

The book gives an answer of y(t)=sqrt(ln|k(t^2+1)|) where k is any real number and the sqrt is either positive or negative.

I understand most of the steps, but why do I get ln|t^2+1| + C inside the square root while the book gets ln|k(t^2+1)| inside the square root?
Define $C=\ln K$.