Dear members, I am unable to solve (dx-dy)/(x-y) = ln(x-y), Please help me in this regard.Thanks
Note: Main goal is to get the x values on one side and the y ones on the other (Seperation of Variables). Next integrate to get the general solution.
So we use multiplication to get rid of denominators. Next, addition and subtraction to separate variables. However, the big problem is the \(\displaystyle \ln\) expression We must get rid of it because the \ln expression contains both the x and y variables which must be separated. We can get rid of it by making it an exponent of e. But in doing so, everything else must also be made an exponent of e. Finally we can do the last x and y separation, convert back to normal expressions taking the ln of everything, integrate and then were done.
\(\displaystyle \dfrac{dx - dy}{x - y} = \ln(x - y)\)
\(\displaystyle (x - y)\dfrac{dx - dy}{x - y} = \ln(x - y) (x - y)\)
\(\displaystyle dx - dy = \ln(x - y) + x - y\)
\(\displaystyle dx - dx - dy = \ln(x - y) + x - y - dx\)
\(\displaystyle -dy = \ln(x - y) + x - y - dx\)
\(\displaystyle -dy + y = \ln(x - y) + x - y + y - dx\)
\(\displaystyle -dy + y = \ln(x - y) + x - dx\)
\(\displaystyle e^{-dy + y} = e^{\ln(x - y) + x - dx}\)
\(\displaystyle e^{-dy} * e^{ y} = x - y * e^{x} * e^{-dx}\)
\(\displaystyle e^{-dy} * e^{ y} - y = x - y - y * e^{x} * e^{-dx}\)
\(\displaystyle e^{-dy} * e^{ y} - y = x * e^{x} * e^{-dx}\)
\(\displaystyle \ln(e^{-dy}) * \ln( e^{ y}) - \ln(y) = \ln(x) * \ln(e^{x}) * \ln(e^{-dx})\)
\(\displaystyle -dy + y - \ln(y) = \ln(x) + x -dx\)
\(\displaystyle (-1)(-dy + y -\ln(y)) = (-1)(\ln(x) + x -dx)\)
\(\displaystyle dy - y + \ln(y) = -\ln(x) - x + dx\)
\(\displaystyle y + \ln(y) + dy = -\ln(x) - x + dx\)
\(\displaystyle \int y + \ln(y) + dy = \int -\ln(x) - x + dx\)
What now? What is the integral of \(\displaystyle \ln\) expression?
