← Engineering Mathematics · UPSC ESE Civil

Chapter 5 of 6

Differential Equations

In the UPSC ESE Civil syllabus under Engineering Mathematics · 2 parts

Ordinary Differential Equations · Partial Differential Equations

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📑 Contents (19 sections)

Part 1 of 2

Ordinary Differential Equations

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 $n$ .

First-order equations

1. Variable separable

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

2. Homogeneous equations

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

3. Exact equations

$M (x, y) d x + N (x, y) d y = 0$ 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:

If $\frac{M _{y} - N _{x}}{N} = f (x)$ → IF $= e^{\int f (x) d x}$ .
If $\frac{N _{x} - M _{y}}{M} = g (y)$ → IF $= e^{\int g (y) d y}$ .

4. Linear first-order equation

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

5. Bernoulli's equation

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

Applications of first-order ODEs

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 $- d x / d y$	—

Higher-order linear equations with constant coefficients

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).

Complementary function

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

Roots	CF
Real and distinct $m_{1}, m_{2}$	$C_{1} e^{m_{1} x} + C_{2} e^{m_{2} x}$
Real and equal $m, m$	$(C_{1} + C_{2} x) e^{m x}$
Complex $α \pm i β$	$e^{α x} (C_{1} cos β x + C_{2} sin β x)$

Particular integral — operator method

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

$F (x)$	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)
$e^{a x} V (x)$	$e^{a x} \frac{1}{f ( D + a )} V$

Method of variation of parameters

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. $sec x$ , $tan x$ ).

Part 2 of 2

Partial Differential Equations

Last reviewed 16 Sept 2026 · 6 min read

Definitions

A partial differential equation (PDE) involves partial derivatives of a function of two or more independent variables.
Notation for $z = z (x, y)$ : $p = \frac{\partial z}{\partial x}$ , $q = \frac{\partial z}{\partial y}$ , $r = \frac{\partial ^{2} z}{\partial x ^{2}}$ , $s = \frac{\partial ^{2} z}{\partial x \partial y}$ , $t = \frac{\partial ^{2} z}{\partial y ^{2}}$ .
Order, degree and linearity defined as for ODEs.

Formation of PDEs

Eliminating arbitrary constants — differentiate partially and eliminate (if the number of constants equals the number of independent variables, a first-order PDE results).
Eliminating arbitrary functions — e.g. $z = f (x + y)$ , $z = f (x^{2} + y^{2})$ .

Lagrange's linear equation

P p + Qq = R (P, Q, R functions of x, y, z)

Auxiliary (subsidiary) equations:

\frac{d x}{P} = \frac{d y}{Q} = \frac{d z}{R}

Find two independent solutions $u (x, y, z) = c_{1}$ and $v (x, y, z) = c_{2}$ ; the general solution is $ϕ (u, v) = 0$ .

(Methods: grouping, and multipliers $l, m, n$ with $l P + m Q + n R = 0$ .)

Classification of second-order linear PDEs

For $A u_{xx} + B u_{x y} + C u_{y y} + (lower-order terms) = 0$ :

Discriminant $B^{2} - 4 A C$	Type	Standard example
$< 0$	Elliptic	Laplace equation $u_{xx} + u_{y y} = 0$ (steady state)
$= 0$	Parabolic	Heat (diffusion) equation $u_{t} = c^{2} u_{xx}$
$> 0$	Hyperbolic	Wave equation $u_{tt} = c^{2} u_{xx}$

Fourier series

A periodic function $f (x)$ of period $2 L$ can be represented as

f (x) = \frac{a _{0}}{2} + n = 1 \sum \infty (a_{n} cos \frac{nπ x}{L} + b_{n} sin \frac{nπ x}{L})

FormulaEuler coefficients (interval −L to L)

a_{0} = \frac{1}{L} \int_{- L}^{L} f (x) d x, a_{n} = \frac{1}{L} \int_{- L}^{L} f (x) cos \frac{nπ x}{L} d x, b_{n} = \frac{1}{L} \int_{- L}^{L} f (x) sin \frac{nπ x}{L} d x

Even function ( $f (- x) = f (x)$ ): $b_{n} = 0$ — cosine series only
Odd function ( $f (- x) = - f (x)$ ): $a_{0} = a_{n} = 0$ — sine series only
Half-range series on $(0, L)$ : sine series $b_{n} = \frac{2}{L} \int_{0}^{L} f (x) sin \frac{nπ x}{L} d x$ ; cosine series $a_{n} = \frac{2}{L} \int_{0}^{L} f (x) cos \frac{nπ x}{L} d x$

Dirichlet conditions — a function that is bounded, has finitely many discontinuities and extrema in a period has a convergent Fourier series; at a jump it converges to the average of left and right limits.

Method of separation of variables

Assume $u (x, t) = X (x) T (t)$ , substitute, separate into ODEs set equal to a separation constant (chosen negative, $- λ^{2}$ , for oscillatory/decaying physically meaningful solutions), solve with boundary conditions, and superpose using Fourier series for initial conditions.

One-dimensional heat (diffusion) equation

\frac{\partial u}{\partial t} = c^{2} \frac{\partial ^{2} u}{\partial x ^{2}}, c^{2} = \frac{k}{ρ s} (diffusivity)

FormulaRod of length L with ends kept at zero temperature

Boundary conditions $u (0, t) = u (L, t) = 0$ ; initial temperature $u (x, 0) = f (x)$ :

u (x, t) = n = 1 \sum \infty b_{n} sin \frac{nπ x}{L} e^{- c^{2} n^{2} π^{2} t / L^{2}}, b_{n} = \frac{2}{L} \int_{0}^{L} f (x) sin \frac{nπ x}{L} d x

Higher modes decay faster; the steady state is $u = 0$ (or linear for fixed unequal end temperatures).

Terzaghi's one-dimensional consolidation equation $\frac{\partial u}{\partial t} = c_{v} \frac{\partial ^{2} u}{\partial z ^{2}}$ (excess pore pressure) has exactly this form (see Consolidation in Geotechnical Engineering).

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