- #1
perishingtardi
- 21
- 1
Suppose I have a variable separable ODE, e.g.,
\frac{dy}{dx} = 3y.
We all know that the solution is y=Ae^{3x} where A is a constant. My question is as follows. To actually find this solution we rearrange the equation and integrate to get
\int \frac{dy}{y} = 3 \int dx,
which gives
\ln |y| = 3x + C where C is a constant. I would have thought that this gives the solution
|y| = Ae^{3x} \qquad \mbox{where} \qquad A=e^C.
My question is how can we get rid of the absolute value sign in the actual answer? Is it because A = e^C must always be positive? But how come in general that does not have to be true for the ODE to be satisfied?
\frac{dy}{dx} = 3y.
We all know that the solution is y=Ae^{3x} where A is a constant. My question is as follows. To actually find this solution we rearrange the equation and integrate to get
\int \frac{dy}{y} = 3 \int dx,
which gives
\ln |y| = 3x + C where C is a constant. I would have thought that this gives the solution
|y| = Ae^{3x} \qquad \mbox{where} \qquad A=e^C.
My question is how can we get rid of the absolute value sign in the actual answer? Is it because A = e^C must always be positive? But how come in general that does not have to be true for the ODE to be satisfied?