Error Analysis Of A Separable Differential Equation

A student’s work for solving the differential equation $\displaystyle \frac{dy}{dx} = \frac{x+1}{y}$ with the initial condition $y(1) = 0$ is shown below. In which step does the first error appear?

Step 1: $\displaystyle \int y dy = \int (x+1) dx$

Step 2: $\displaystyle \frac{y^2}{2} = \frac{x^2}{2} + x + C$

Step 3: Using the initial condition: $\displaystyle \frac{0^2}{2} = \frac{1^2}{2} + 1 + C$

Step 4: $\displaystyle 0 = \frac{1}{2} + 1 + C$, so $C = -\frac{3}{2}$

Step 5: $\displaystyle \frac{y^2}{2} = \frac{x^2}{2} + x - \frac{3}{2}$

Step 6: $y^2 = x^2 + 2x - 3$

Step 7: $y^2 = (x+1)^2 - 4$

Step 8: $y = \sqrt{(x+1)^2 - 4}$