Solution to Differential Equation/Examples/Equation which is Not a Solution

Examples of Solutions to Differential Equations
Consider the equation:


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

where $x \in \R$.

Consider the first order ODE:
 * $(2): \quad x + y y' = 0$

Then despite the fact that the formal substition for $y$ and $y'$ from $(1)$ into $(2)$ yields an identity, $(1)$ is not a solution to $(2)$.

Proof
First we note that:

Then:

However, by Domain of Real Square Root Function, $\sqrt {-\paren {1 + x^2} }$ is defined for $-\paren {1 + x^2} \ge 0$.

But by Square of Real Number is Non-Negative:
 * $1 + x^2 > 0$

and so:
 * $-\paren {1 + x^2} < 0$

So there exists no $x \in \R$ for which $y = \sqrt {-\paren {1 + x^2} }$ is defined.

So $(1)$ does not define a real function and so $(1)$ is not a solution to $(2)$.