Ordinary Differential Equations Notes

Last reviewed 16 Sept 2026 · 7 min read

Basic definitions

Ordinary differential equation (ODE) — involves derivatives with respect to one independent variable.
Order — highest derivative present.
Degree — power of the highest-order derivative after removing radicals and fractions in derivatives.
Linear — dependent variable and its derivatives appear to the first power and are not multiplied together.
General solution contains as many arbitrary constants as the order; particular solution — constants fixed by initial/boundary conditions.
Formation — eliminating $n$ arbitrary constants gives an ODE of order $n$ .

$f (x) d x = g (y) d y$ → integrate both sides.

$\frac{d y}{d x} = F (\frac{y}{x})$ — substitute $y = v x$ , $\frac{d y}{d x} = v + x \frac{d v}{d x}$ → separable.

$M (x, y) d x + N (x, y) d y = 0$ is exact if

\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}

Solution: $\int M d x (y constant) + \int (terms of N not containing x) d y = C$ .

Integrating factors for non-exact equations:

FormulaLinear equation

\frac{d y}{d x} + P (x) y = Q (x)

Integrating factor $I F = e^{\int P d x}$ ; solution

y \cdot I F = \int Q \cdot I F d x + C

$\frac{d y}{d x} + P y = Q y^{n}$ — divide by $y^{n}$ and substitute $z = y^{1 - n}$ → linear in $z$ .

Application	Model	Solution
Exponential growth/decay (population, radioactive decay, compound interest)	$\frac{d N}{d t} = k N$	$N = N_{0} e^{k t}$ ; half-life $t_{1/2} = \frac{ln 2}{∣ k ∣}$
Newton's law of cooling	$\frac{d T}{d t} = - k (T - T_{s})$	$T = T_{s} + (T_{0} - T_{s}) e^{- k t}$
RC circuit (charging)	$R C \frac{d q}{d t} + q = C V$	$q = C V (1 - e^{- t / R C})$ ; time constant $R C$
RL circuit	$L \frac{d i}{d t} + R i = V$	$i = \frac{V}{R} (1 - e^{- R t / L})$
Mixing problems	Rate in − rate out	Linear ODE
Orthogonal trajectories	Replace $d y / d x$ by $- d x / d y$	—

a_{0} \frac{d ^{n} y}{d x ^{n}} + a_{1} \frac{d ^{n - 1} y}{d x ^{n - 1}} + \dots + a_{n} y = F (x)

General solution = complementary function (CF) + particular integral (PI).

Solve the auxiliary equation (replace $D = d / d x$ by $m$ ), e.g. $a m^{2} + bm + c = 0$ :

$P I = \frac{1}{f ( D )} F (x)$ :

$F (x)$	Rule
$e^{a x}$	$\frac{e ^{a x}}{f ( a )}$ if $f (a) \neq = 0$ ; if $f (a) = 0$ (resonance), $x \frac{e ^{a x}}{f ^{'} ( a )}$
$sin a x$ or $cos a x$	Replace $D^{2}$ by $- a^{2}$ in $f (D)$ ; if denominator becomes zero, multiply by $x$ and differentiate the denominator — e.g. $\frac{1}{D ^{2} + a ^{2}} sin a x = - \frac{x}{2 a} cos a x$
$x^{n}$ (polynomial)	Expand $[f (D)]^{- 1}$ in ascending powers of $D$ (binomial series)
$e^{a x} V (x)$	$e^{a x} \frac{1}{f ( D + a )} V$

For $y^{''} + p (x) y^{'} + q (x) y = r (x)$ with CF $C_{1} y_{1} + C_{2} y_{2}$ :

P I = - y_{1} \int \frac{y _{2} r}{W} d x + y_{2} \int \frac{y _{1} r}{W} d x, W = y_{1} y_{2}^{'} - y_{2} y_{1}^{'}

(Wronskian $W$ .) Works for any $r (x)$ (e.g. $sec x$ , $tan x$ ).

Downloaded from civilgyantests.com · Not for resale. Verify every code clause and date against the official source.