Separation of Variables/Examples/Arbitrary Example 1

Example of Use of Separation of Variables
Consider the first order ODE:
 * $(1): \quad \map {\dfrac \d {\d x} } {\map f x} = 3 x$

where we are given that $\map f 1 = 2$.

The particular solution to $(1)$ is:
 * $\map f x = \dfrac {3 x^2 + 1} 2$

Proof
We use the technique of Separation of Variables.

When $x = 1$, we have that $\map f x = 2$.

Hence:

Hence:
 * $\map f x = \dfrac {3 x^2 + 1} 2$