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