Error Analysis in Solving a Differential Equation

A student attempted to solve the differential equation $\displaystyle \frac{dy}{dx} = \frac{1-y}{x}$ with the initial condition that $y=0$ when $x=1$. The student’s work is shown below. In which step does an error first appear?

Step 1: $\displaystyle \int \frac{1}{1-y} dy = \int \frac{1}{x} dx$

Step 2: $-\ln |1-y| = \ln |x| + C$

Step 3: $\ln |1-y| = -\ln |x| - C$

Step 4: $\ln |1-y| = \ln \left|\frac{1}{x}\right| - C$

Step 5: $|1-y| = \frac{K}{x}$ where $K = e^{-C}$

Step 6: Using the initial condition: $|1-0| = \frac{K}{1}$, so $K = 1$

Step 7: $|1-y| = \frac{1}{x}$

Step 8: $1-y = \pm\frac{1}{x}$

Step 9: Since $y = 0$ when $x = 1$, we have $1-y = \frac{1}{x}$

Step 10: $y = 1 + \frac{1}{x}$