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Formula Sheets / Maths

Chapter 22

Differential Equations

Revise with flashcards Download this chapter (PDF) All Maths formulas (PDF)

Order, degree and formation Variable separable and homogeneous Linear differential equations Applications

Order, degree and formation

order $=$ highest derivative $;$ degree $=$ its power, once the equation is polynomial in derivatives
Formation
$n$ arbitrary constants $\Rightarrow$ differentiate $n$ times and eliminate them (order $n)$

Variable separable and homogeneous

Separable
$f (y) d y = g (x) d x \Rightarrow \int f (y) d y = \int g (x) d x + C$
Homogeneous
$\frac{d y}{d x} = F (\frac{y}{x}) : y = v x, \frac{d y}{d x} = v + x \frac{d v}{d x}$

Linear differential equations

$\frac{d y}{d x} + P (x) y = Q (x) :$ IF $= e^{\int P d x},$ $y \cdot$ IF $= \int Q \cdot$ IF $d x + C$
Bernoulli
$\frac{d y}{d x} + P y = Q y^{n} :$ put $z = y^{1 - n}$
Exact forms
$x d y + y d x = d (x y),$ $\frac{x d y - y d x}{x ^{2}} = d (\frac{y}{x}),$ $\frac{x d y - y d x}{x ^{2} + y ^{2}} = d (tan^{- 1} \frac{y}{x})$

Applications

Growth and decay
$\frac{d N}{d t} = k N \Rightarrow N = N_{0} e^{k t}$
Newton's law of cooling
$\frac{d T}{d t} = - k (T - T_{s})$

Area Under Curves Straight Lines