Simplest Ordinary Differential Equation:

\[ \begin{aligned} \frac{dx}{dt} = ax \text{ , then } x &= Ce^{at} \end{aligned} \]

Linear 1st order differential equation:

\[ \begin{aligned} \frac{dx}{dt} = A(t)x + B(t) \end{aligned} \]

Standard form:

\[ \begin{aligned} \frac{dx}{dt} - Ax = B \end{aligned} \]

Integrating factor:

Used for solving linear 1st order differential equations via the product rule to factor the sum of two derivatives into the derivative of a product.

\[ \begin{aligned} I &= e^{\int{-A}} \\ \text{i.e. } I &= e^{\int{-A(t)dt}} \end{aligned} \]

such that:

\[ \begin{aligned} I\frac{dx}{dt} - IAx &= IB = \frac{d}{dt}Ix \\ \end{aligned} \]

Solution:

\[ \begin{aligned} x &= e^{\int{A}} \cdot \int{B e^{\int{-A}}} \\ \text{i.e. } x &= e^{\int{A(t)dt}} \cdot \int{B(t) e^{\int{-A(t)dt}}} \\ \end{aligned} \]

Equilibrium of \(\displaystyle \frac{dx}{dt} = f(x)\)

are simply the roots of \(f(x)\). Note if \(f'(x) < 0\), the equilibrium is stable. If \(f'(x) > 0\), the equilibrium is unstable.

Partial Fractions

\[ \begin{aligned} \frac{P(x)}{Q(x)} = \frac{A_1}{x-r_1} + \frac{A_2}{x-r_2} + \cdots + \frac{A_n}{x-r_n} \end{aligned} \]

Fundamental Theorem of Integral Calculus

\[ \begin{aligned} \frac{d}{dx} \int_{t=a}^x f(t)dt = f(x) \end{aligned} \]

p-integral

\( \displaystyle \int \frac{1}{x^p} dx = \int x^{-p} dx \)

Note

\(\displaystyle \int_1^\infty x^{-p}dx \)

diverges if \(p \le 1\) and converges otherwise; whereas

\(\displaystyle \int_0^1 x^{-p}dx \)

diverges if \(p \ge 1\) and converges otherwise.

For more info, check out the PennCalcWiki.