Separation of Variables Error Analysis

The following steps show a student’s attempt to solve the differential equation $\displaystyle \frac{dy}{dx} = y \sin x$ with the initial condition that $y = 1$ when $x = 0$. In which step does an error first appear?

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

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

Step 3: Using the initial condition: $\ln |1| = -\cos(0) + C$, so $C = 1$

Step 4: $\ln |y| = -\cos x + 1$

Step 5: $|y| = e^{-\cos x + 1}$

Step 6: $y = e \cdot e^{-\cos x}$

Step 7: $y = e^{1-\cos x}$

Step 8: $y = e^{\cos x - 1}$