Solution to Differential Equation/Examples/Arbitrary Order 1 ODE: 2

Examples of Solutions to Differential Equations
Consider the real function defined as:


 * $y = \tan x - x$

defined on the domain $S := \set {x \in \R: x \ne \dfrac {\paren {2 n + 1} \pi} 2, n \in \Z}$.

Then $\map f x$ is a solution to the first order ODE:
 * $(1): y' = \paren {x + y}^2$

when $x$ is restricted to $S$.

Proof
It is noted that $\map f x$ is not defined in $\R$ when $x = \dfrac {\paren {2 n + 1} \pi} 2$ because for those values of $x$ the tangent is not defined.

Hence the restriction of $x$ to $S$

It is also noted that $(1)$ is indeed defined for all $x \in S$.

Having established that, we continue:

For ease of manipulation we rewrite $(1)$ as:


 * $(2): y' - \paren {x + y}^2 = 0$

Then: