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## Fourier Series

Let $f$ be a piecewise continuous function on the interval $[-T,T]$.  The Fourier Series of $f$ is the trigonometric series

where the $a_n$'s and $b_n$'s are given by the formulas

For example, let $f(x)$ be defined by !f(x)=\begin{cases} 0 & , […]

## Fun with Trigonometry

The homogeneous solution is

Using variation of parameters we get a nonhomogeneous solution of

Distributing and regrouping

Reversing the cosine angle difference formula

Using the initial conditions $x(0)=0$, $\dot{x}(0)=0$, we get $c_1=\frac{-25}{18}$ and $c_2=0$ So a specific solution is

Regrouping and applying a difference to product formula

[…]

## Bernoulli Equations

Bernoulli equations are of the form:

Dividing by $y^n$:

If $n=0$ or $n=1$ then the equation is linear.  Otherwise a substitution of $u=y^{1-n}$ will give us a linear equation. If $u=y^{1-n}$, then $\frac{du}{dx}u=(1-n)\frac{dy}{dx}y^{-n}$.  Substituting these into the second equation from above:

For example,

Dividing by $y^4$, we get  $y'y^{-4}+2xy^{-3}=x$ Substituting $u=y^{-3}$ […]

## Homogeneous First-order Equations

There are two ways to view "homogeneous" first-order ordinary differential equations. 1. $\frac{dy}{dx}=f(x,y)$ is a general first order ordinary differential equation.  If the function on the right, $f(x,y)$, can be rewritten as a function of $\frac{y}{x}$, e.g. $f(x,y)=\frac{y-2x}{x}$ can be rewritten as $g\left(\frac{y}{x}\right)=\frac{y}{x}-2$.  If we let $v=\frac{y}{x}$, then we would have $g(v)=v-2$. In general, the […]