What is the correct answer to this diffferential equation?

Question

[Question] When solving the differential equation:

$$\frac{\mathrm dy}{\mathrm dx} = \sqrt{(y+1)}$$

I've found two ways to express $y(x)$:

implicitly: $2\sqrt{(y + 1)} = x + C$

or directly: $y = (x^2)/4 + (2xC)/4 + (C^2)/4 -1$

Although they look the same, these expressions result in different answers when applying the initial condition $y(0) =1$:

When using the implicit expression:

$$2\sqrt{(y(0) + 1)} = 0 + C \\ 2\sqrt{(1 + 1)} = 0 + C \\ C = 2\sqrt{2}$$

When using the direct expression:

$$y(0) = 0/4 + 0/4 + (C^2)/4 -1 \\ 2 = (C^2)/4 \\ C = \pm 2\sqrt{2} \\ \text{since: } C^2 = 8 \implies C = \pm \sqrt 8 $$

Thus, using the direct expression results in two answers (being a quadratic equation), when using the implicit expression there's only one answer to C.

Which one is the complete answer?

[Additional information] The solution's manual states that the answer should be:

$C = 2\sqrt 2 $ and the solution to the differential equation with initial condition is $2\sqrt{(y + 1)} = x + 2\sqrt 2$. (notice, no minus sign before $2\sqrt 2 $)

However, I do not think this is the complete solution, as shown by the direct expression, there may be another answer to the differential equation:

$C = - 2\sqrt(2)$ and the solution is: $2\sqrt{(y + 1)} = x - 2\sqrt 2$

I do not know if the implicit expression hides one of the answers or if is there something wrong with my use of the direct expression.

Answer 1

The version with $\sqrt{8}$ is the only correct one. If you use $-\sqrt{8}$, you get that $\sqrt{y+1}$ is negative when $x=0$. But $\sqrt{y+1}$, by definition, is the non-negative number whose square is $y+1$.

Remark: A familiar related fact is that when we are solving an ordinary equation that involves square roots, squaring often introduces one or more extraneous solutions, that is, non-solutions.

Answer 2

Your explicit solution is $$y=\Bigl(\frac{x+C}{2}\Bigr)^2-1\ .$$ This is equivalent to $$x+C=\pm2\sqrt{y+1}\ ,$$ not $x+C=2\sqrt{y+1}$ as it is supposed to be. And indeed, with the minus sign you can differentiate with respect to $x$ to get $$1=-\frac1{\sqrt{y+1}}\frac{dy}{dx}\ ,$$ that is, $$\frac{dy}{dx}=-\sqrt{y+1}\ ,$$ which is not the given DE. So only the plus sign is correct. This means that your explicit solution includes the correct solution and an incorrect solution too.