While solving differential equations I noticed that I repeatedly made one specific error which can best be demonstrated with this simple example:
[math]y' = 1-y[/math]
Seperation of variables:
[math]\frac{dy}{1-y} = dx[/math]
[math]\implies \int \frac{dy}{1-y} = \int dx[/math]
u-sub with [imath]u = y -1[/imath] for the left integral gives
[math]\log{|1-y|} = -x + C[/math]
Now here is where I made my error, when taking the exponential function of both sides, this gives
[imath]y-1=\tilde{C}e^{-x} \space [/imath] and NOT [imath] \space 1-y=\tilde{C}e^{-x}[/imath]
I understand we need the absolute value sign for the argument of log so we don't take the log of a negative number. But how does the absolute value vanish when taking the exponential function? And then - why do we need to switch from [imath]\log{|1-y|}[/imath] to [imath]\log{|y-1|}[/imath] ?
[math]y' = 1-y[/math]
Seperation of variables:
[math]\frac{dy}{1-y} = dx[/math]
[math]\implies \int \frac{dy}{1-y} = \int dx[/math]
u-sub with [imath]u = y -1[/imath] for the left integral gives
[math]\log{|1-y|} = -x + C[/math]
Now here is where I made my error, when taking the exponential function of both sides, this gives
[imath]y-1=\tilde{C}e^{-x} \space [/imath] and NOT [imath] \space 1-y=\tilde{C}e^{-x}[/imath]
I understand we need the absolute value sign for the argument of log so we don't take the log of a negative number. But how does the absolute value vanish when taking the exponential function? And then - why do we need to switch from [imath]\log{|1-y|}[/imath] to [imath]\log{|y-1|}[/imath] ?
Last edited: