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First Order Linear ODE
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Summary
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A Linear Odinary Differential Equation is of the form:

$$\begin{aligned}
\sum^{n}_{0} \left[ a_n\left( x \right)\cdot f^{\left( n \right)} \left( x \right)\right] = g\left( x \right)&\\
&\text{If $g\left( x \right)= 0$ it is said to be homogenous}\end{aligned}$$

A first Order Linear ODE is of the form:

$$\begin{aligned}
a_1\left( x \right)\cdot \frac{\operatorname{d}y }{\operatorname{d} x}+ a_0\left( x \right)\cdot y= g\left( x \right)\notag &\\
&\text{Where $a\left( x \right)$ is a function } \end{aligned}$$

It is typical to rewrite this as:   

**Linear First Order ODE:** $$\begin{aligned}
\frac{\operatorname{d}y }{\operatorname{d} x} + p\left( x \right)\cdot y = f\left( x \right)\end{aligned}$$
if $f\left( x \right)= 0$ the equation is said to be homogenous

Let the homogenous be $y_h$ and the particular solution be $y_p$, i.e.:

- $y_h: \quad \frac{\operatorname{d}y_h }{\operatorname{d} x}+ p\left( x \right)\cdot y_h= 0$

- $y_p: \quad \frac{\operatorname{d}y_p }{\operatorname{d} x}+ p\left( x \right)\cdot y_p= f\left( x \right)$

In order to find a solution a solution for a First Order Linear ODE,
don't remember an equation, remember the technique:

- Rewrite the Equation in the standard form:
$$\frac{\operatorname{d}y }{\operatorname{d} x}+ p\left( x \right)\cdoty = f \left( x \right)$$

- Identify $p\left( x \right)$ and find the integrating factor:
$$e^{\int^{}_{} p\left( x \right) \operatorname{d}x }$$

- Multiply through by the integrating factor:
$$e^{\int^{}_{} p\left( x \right) \operatorname{d}x } \left( \frac{\operatorname{d}y }{\operatorname{d} x}+ p\left( x \right)\cdoty \right) = e^{\int^{}_{} p\left( x \right) \operatorname{d}x }f \left( x \right)$$

It may be concluded:   

$$\frac{\operatorname{d} }{\operatorname{d} x}\left[ e^{\int^{}_{} p\left( x \right) \operatorname{d}x \cdot } \cdot y\right] = e^{\int^{}_{} p\left( x \right) \operatorname{d}x } \cdot f \left( x \right)$$

- Integrate both sides in order to solve:

Proof
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Exemplars
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