CELE Engineering Mathematics — Differential EquationsCheat Sheet

Formula

Order = highest derivative present; Degree = power of highest-order derivative (after polynomial form)

Meaning

Order identifies the number of integration constants in general solution; Degree tells the highest power when DE is polynomial in derivatives

Watch Out

Degree applies only AFTER clearing radicals and fractions in derivatives; a DE may be nonlinear even if first-order

When To Use

Always identify order and degree FIRST before selecting solution method

Term

Ordinary Differential Equation (ODE)

Example

dy/dx + 2y = x² (one independent variable x)

Definition

Equation relating a function of one independent variable to its derivatives; opposed to partial DEs with multiple independent variables.

Term

Linear ODE

Example

y'' + 3y' + 2y = 5 is linear; (y')² + y = 0 is nonlinear

Definition

The dependent variable y and all its derivatives appear to the first power only, with no products of y and its derivatives.

Term

Homogeneous ODE

Example

y' = 2y is homogeneous; y' = 2y + 3 is non-homogeneous

Definition

Right-hand side equals zero; e.g., y'' + 5y' + 6y = 0 (no forcing term).

Term

Particular Solution

Example

y = x² + 1 satisfying y(0) = 1

Definition

A specific solution satisfying given initial or boundary conditions; contains no arbitrary constants.

Term

General Solution

Example

y = C₁e^(2x) + C₂e^(3x) for a second-order ODE

Definition

Contains arbitrary constants (number = order of DE); represents all possible solutions.

Formula

Separable: dy/dx = g(x)h(y) ⟹ ∫[dy/h(y)] = ∫g(x)dx

Meaning

g(x) depends on x only; h(y) depends on y only; rearrange so all y terms on left, all x terms on right

Watch Out

DO NOT forget the constant of integration; do NOT cancel h(y) if it equals zero (singular solution may exist)

When To Use

When the RHS factors into a product of a function of x and a function of y

Formula

Linear first-order: dy/dx + P(x)y = Q(x); Integrating Factor μ = e^(∫P(x)dx)

Meaning

P(x) is coefficient of y; Q(x) is forcing term; multiply entire equation by μ

Watch Out

Sign in P(x) must match the ODE form EXACTLY; integrate P(x) carefully; don't add constant when finding μ

When To Use

Standard form dy/dx + P(x)y = Q(x); works for ALL such equations regardless of whether separable

Formula

After multiplying by μ: d/dx[μy] = Q(x)μ ⟹ μy = ∫Q(x)μ(x)dx

Meaning

LHS becomes exact derivative of product μy; integrate RHS to find y

Watch Out

The derivative d/dx[μy] = μ(dy/dx) + y(dμ/dx) = μ(dy/dx) + yP(x)μ — verify the product rule carefully

When To Use

Always the step after multiplying first-order linear ODE by integrating factor

Formula

Exact equation: M(x,y)dx + N(x,y)dy = 0 with ∂M/∂y = ∂N/∂x

Meaning

M is coefficient of dx; N is coefficient of dy; partial derivatives must be equal

Watch Out

ALWAYS verify ∂M/∂y = ∂N/∂x BEFORE proceeding; if not exact, may need integrating factor

When To Use

Check exactness first; if exact, solve by finding F(x,y) such that ∂F/∂x = M and ∂F/∂y = N

Formula

Solution to exact equation: F(x,y) = C where ∂F/∂x = M and ∂F/∂y = N

Meaning

Find F by integrating M w.r.t. x (treating y as constant), then integrate N w.r.t. y and match terms

Watch Out

When integrating M w.r.t. x, the arbitrary 'constant' is a function of y; use the N equation to determine it

When To Use

After confirming exactness; F is the implicit solution

Term

Integrating Factor (IF)

Example

For dy/dx + 2y = x, μ = e^(∫2dx) = e^(2x)

Definition

Function μ(x) that, when multiplied by an ODE, converts it to exact or allows direct integration; μ = e^(∫P(x)dx) for linear first-order

Term

Exact Equation

Example

(2x + y)dx + (x + 2y)dy = 0 is exact since ∂M/∂y = 1 = ∂N/∂x

Definition

Differential equation M dx + N dy = 0 where ∂M/∂y = ∂N/∂x; solution is F(x,y) = C (implicit form)

Term

Singular Solution

Example

For dy/dx = y², general solution is y = -1/(x+C); singular solution is y = 0

Definition

A solution not obtainable from the general solution for ANY value of the arbitrary constant; may arise from separable equations where h(y) = 0

Formula

General form: aₙ(d^n y/dx^n) + aₙ₋₁(d^(n-1) y/dx^(n-1)) + ⋯ + a₁(dy/dx) + a₀y = f(x)

Meaning

aᵢ are constant coefficients; y is dependent variable; f(x) is forcing function

Watch Out

If f(x) = 0, it's homogeneous; if f(x) ≠ 0, it's non-homogeneous; each requires different approach

When To Use

This is the canonical form for all linear constant-coefficient ODEs

Formula

Characteristic equation (homogeneous case f(x)=0): aₘ² + bm + c = 0 for ay'' + by' + cy = 0

Meaning

Replace y with e^(mx), y' with me^(mx), y'' with m²e^(mx); divide by e^(mx) to get characteristic equation

Watch Out

Do NOT forget to form the characteristic equation for each term; roots determine solution form

When To Use

ALWAYS for homogeneous linear constant-coefficient ODEs

Formula

Distinct real roots m₁, m₂: y = C₁e^(m₁x) + C₂e^(m₂x)

Meaning

C₁ and C₂ are arbitrary constants determined by initial conditions

Watch Out

If roots are negative, solution decays to zero; if positive, exponential growth

When To Use

When characteristic equation has two different real roots

Formula

Repeated real root m (multiplicity 2): y = (C₁ + C₂x)e^(mx)

Meaning

Second solution includes a factor of x to ensure linear independence; C₁ and C₂ are constants

Watch Out

CANNOT use y = C₁e^(mx) + C₂e^(mx) for repeated roots — you get linearly dependent solutions

When To Use

When characteristic equation has one repeated root (discriminant = 0)

Formula

Repeated root m (multiplicity k): y = (C₁ + C₂x + C₃x² + ⋯ + Cₖx^(k-1))e^(mx)

Meaning

For root of multiplicity k, include polynomial of degree k-1 as coefficient of e^(mx)

Watch Out

Number of terms = multiplicity; powers of x go from 0 to k-1

When To Use

When characteristic equation has a root with multiplicity k ≥ 2

Formula

Complex conjugate roots α ± βi: y = e^(αx)(C₁cos(βx) + C₂sin(βx))

Meaning

α is real part (damping); β is imaginary part (frequency); C₁ and C₂ determined by ICs

Watch Out

Do NOT use e^(αx)·e^(±βix) — convert to real trigonometric form using Euler's formula

When To Use

When characteristic equation yields complex roots

Formula

Superposition for homogeneous: If y₁ and y₂ are solutions, then y = C₁y₁ + C₂y₂ is also a solution

Meaning

Arbitrary linear combinations of solutions are solutions (applies to LINEAR homogeneous ODEs only)

Watch Out

Does NOT work for nonlinear ODEs

When To Use

Always; foundational property for building general solutions

Formula

Non-homogeneous solution: y = yₕ + yₚ where yₕ = complementary function, yₚ = particular solution

Meaning

yₕ solves the homogeneous part (ay'' + by' + cy = 0); yₚ satisfies the full non-homogeneous equation

Watch Out

Find yₕ first (using characteristic equation); then find yₚ using method of undetermined coefficients or variation of parameters

When To Use

ALWAYS for non-homogeneous constant-coefficient ODEs

Formula

Method of Undetermined Coefficients — Assume form for yₚ based on f(x); assume polynomial, exponential, or sinusoidal depending on f(x)

Meaning

Guess a form with unknown coefficients; substitute into ODE; solve for coefficients

Watch Out

If your assumed form for yₚ is a solution to the homogeneous ODE, multiply by x (or x² if repeated); avoid redundancy with yₕ

When To Use

When f(x) is polynomial, exponential, or sinusoidal (or sum/product thereof)

Formula

Particular solution via Undetermined Coefficients — Common guesses: f(x) = Pₙ(x) ⟹ yₚ = Axⁿ+Bxⁿ⁻¹+⋯; f(x) = e^(ax) ⟹ yₚ = Ae^(ax); f(x) = sin(bx) or cos(bx) ⟹ yₚ = Asin(bx)+Bcos(bx)

Meaning

Match the functional form of f(x) when guessing yₚ

Watch Out

If your guess matches part of yₕ, multiply entire yₚ by x; e.g., if yₕ contains e^(2x) and f(x) = e^(2x), use yₚ = Axe^(2x)

When To Use

Standard method for constant-coefficient non-homogeneous ODEs

Formula

Variation of Parameters (for any f(x), not just standard forms): yₚ = -y₁∫(y₂f/W)dx + y₂∫(y₁f/W)dx where W = y₁y₂' - y₂y₁' (Wronskian)

Meaning

y₁ and y₂ are two linearly independent solutions of homogeneous ODE; W is determinant of their Wronskian matrix

Watch Out

Wronskian W ≠ 0 for linearly independent solutions; can be tedious, but universally applicable

When To Use

When f(x) is not amenable to undetermined coefficients; works for ANY f(x)

Term

Characteristic Equation

Example

For y'' - 5y' + 6y = 0, characteristic equation is m² - 5m + 6 = 0 with roots m = 2, 3

Definition

Polynomial equation obtained by substituting y = e^(mx) into homogeneous linear constant-coefficient ODE; roots determine form of solution

Term

Complementary Function (yₕ)

Example

For y'' + y' = x, yₕ = C₁ + C₂e^(-x) (solution to y'' + y' = 0)

Definition

General solution to the homogeneous part of a non-homogeneous ODE; contains all arbitrary constants

Term

Particular Solution (yₚ)

Example

For y'' + y' = x, yₚ = x² - 2x is a particular solution

Definition

Any single solution to the non-homogeneous ODE; contains NO arbitrary constants

Term

Wronskian (W)

Example

For y₁ = e^(2x) and y₂ = e^(3x), W = e^(2x)·3e^(3x) - e^(3x)·2e^(2x) = e^(5x) ≠ 0

Definition

Determinant W = y₁y₂' - y₂y₁' used to check linear independence and in variation of parameters; W ≠ 0 iff y₁ and y₂ are linearly independent

Term

Linear Independence

Example

e^(2x) and e^(3x) are linearly independent; e^(2x) and 2e^(2x) are linearly dependent

Definition

Two solutions y₁ and y₂ are linearly independent if no constant k makes y₁ = ky₂; essential for a valid general solution

Formula

Exponential Growth/Decay: dy/dt = ky ⟹ y(t) = y₀e^(kt)

Meaning

y₀ is initial value; k is growth rate (k > 0) or decay rate (k < 0); t is time

Watch Out

Sign of k: positive = growth; negative = decay; find k from half-life or doubling time data

When To Use

Population growth, radioactive decay, bacterial culture, compound interest

Formula

Radioactive Decay Half-Life: t₁/₂ = ln(2)/|k| where y(t) = y₀e^(-|k|t)

Meaning

Time required for quantity to reduce to half; relates decay constant to observable half-life

Watch Out

Use ln(2) ≈ 0.693; half-life is POSITIVE even though k is negative

When To Use

When given half-life and asked to find amount remaining at a future time

Formula

Newton's Law of Cooling: dT/dt = -k(T - Tₛ) ⟹ T(t) = Tₛ + (T₀ - Tₛ)e^(-kt)

Meaning

T = temperature at time t; T₀ = initial temperature; Tₛ = surrounding temperature; k = cooling constant (k > 0)

Watch Out

Rate is proportional to temperature DIFFERENCE, not absolute temperature; ambient temperature Tₛ is constant

When To Use

Heat transfer, cooling of hot objects in ambient environment

Formula

Mixing Tanks (Brine/Solute): rate of change of solute = (rate in) - (rate out); input conc. × flow in - output conc. × flow out

Meaning

dQ/dt = C_in·F_in - C_out·F_out where Q = amount of solute, C = concentration, F = flow rate

Watch Out

Output concentration = Q(t)/V(t) where V(t) is volume at time t; volume may change if inflow ≠ outflow

When To Use

Tank mixing problems with inflow/outflow of solution at different concentrations

Formula

Mixing with Constant Volume: dQ/dt = (C_in·F) - (Q/V)·F ⟹ Linear first-order ODE in Q(t)

Meaning

When inflow = outflow, tank volume V is constant; C_in is concentration of inflow; F is volumetric flow rate

Watch Out

Output concentration is Q(t)/V because total amount Q is distributed in volume V

When To Use

Standard tank mixing setup (water in = water out)

Formula

Motion with Damping: m(d²x/dt²) + c(dx/dt) + kx = F(t)

Meaning

m = mass; c = damping coefficient; k = spring stiffness; F(t) = external force; x = displacement

Watch Out

Second-order ODE; damping term c(dx/dt) opposes motion; underdamped (c² < 4mk) gives oscillation

When To Use

Vibration analysis, spring-mass-damper systems in mechanics

Formula

Free Vibration (F(t)=0): Characteristic equation: mλ² + cλ + k = 0; discriminant Δ = c² - 4mk determines behavior

Meaning

λ = (−c ± √(c²−4mk))/(2m); Δ > 0 = overdamped; Δ = 0 = critically damped; Δ < 0 = underdamped (oscillatory)

Watch Out

Only Δ < 0 (underdamped) produces oscillations; overdamped and critically damped approach equilibrium without oscillating

When To Use

Predicting damping behavior of mechanical systems (no external forcing)

Value

0.693

Symbol

ln(2)

Quantity

Natural logarithm of 2 (for half-life conversion)

Value

1.0

Symbol

ln(e)

Quantity

Natural logarithm of e

Value

1/e ≈ 0.368 or 36.8%

Symbol

e^(-1)

Quantity

Fraction remaining after one time constant τ

Term

Decay Constant (k)

Example

For Cobalt-60 with t₁/₂ = 5.27 years, k = 0.693/5.27 ≈ 0.131 per year

Definition

Proportionality constant in dy/dt = -ky for decay processes; related to half-life by t₁/₂ = ln(2)/k

Term

Damping Ratio (ζ)

Example

For m = 1 kg, c = 2 N·s/m, k = 1 N/m: ζ = 2/(2√1) = 1 (critically damped)

Definition

Dimensionless parameter ζ = c/(2√(mk)) that classifies oscillatory behavior; ζ < 1 = underdamped (oscillation); ζ = 1 = critically damped; ζ > 1 = overdamped

Term

Time Constant (τ)

Example

RC circuit: τ = RC; for R = 1 kΩ, C = 1 μF, τ = 1 ms

Definition

Characteristic time scale of exponential decay/growth; for dy/dt = -ky, τ = 1/k; after time τ, quantity reduces to 1/e ≈ 37% of initial

Formula

Laplace Transform definition: L{f(t)} = F(s) = ∫₀^∞ e^(-st)f(t)dt

Meaning

Converts function f(t) in time domain to F(s) in frequency domain; s is complex variable

Watch Out

Integral must converge (requires appropriate growth conditions on f); s must be large enough

When To Use

Transform ODE with initial conditions into algebraic equation; solves linear constant-coefficient ODEs with ICs

Formula

Linearity: L{af(t) + bg(t)} = aF(s) + bG(s)

Meaning

Transform of sum = sum of transforms; scaling is preserved

Watch Out

Works because integration is linear

When To Use

Apply to each term of ODE separately

Formula

First derivative: L{f'(t)} = sF(s) - f(0)

Meaning

Derivative in time domain ↔ multiplication by s minus initial condition in transform domain

Watch Out

Do NOT forget the initial condition −f(0); it becomes part of the algebraic equation

When To Use

Replace each y' with sY(s) - y(0) when transforming ODE

Formula

Second derivative: L{f''(t)} = s²F(s) - sf(0) - f'(0)

Meaning

Second derivative ↔ s² times transform minus initial value minus initial slope

Watch Out

Both y(0) AND y'(0) appear; order matters

When To Use

For second-order ODEs; replace each y'' with s²Y(s) - sy(0) - y'(0)

Formula

Heaviside step function: L{u(t-a)} = e^(-as)/s where u(t-a) = 0 for t < a, 1 for t ≥ a

Meaning

Shifted unit step; useful for modeling on/off forcing

Watch Out

Shift in time domain ↔ exponential factor e^(-as) in transform domain

When To Use

Piecewise forcing functions or delayed responses

Formula

Common transforms (memorize): L{1} = 1/s; L{e^(at)} = 1/(s-a); L{sin(bt)} = b/(s²+b²); L{cos(bt)} = s/(s²+b²); L{tⁿ} = n!/s^(n+1)

Meaning

Table of standard function pairs; used for forward and inverse transforms

Watch Out

Memorize at least 5–6 most common transforms; know the shifting theorem for translations

When To Use

Look up or recall when setting up transform equations and performing partial fraction decomposition

Formula

Partial Fraction Decomposition (for inverse transform): Expand F(s) = P(s)/Q(s) as sum of simpler fractions; identify each term with standard transform

Meaning

Break complex rational function into sum of simple fractions (each corresponding to a known time-domain function)

Watch Out

Degree of P < degree of Q required (if not, perform polynomial long division first); each pole contributes a residue term

When To Use

After solving the transformed algebraic equation for Y(s), decompose to find y(t) = L⁻¹{Y(s)}

Formula

Convolution theorem: L{f(t) * g(t)} = F(s)G(s) where f(t) * g(t) = ∫₀^t f(τ)g(t−τ)dτ

Meaning

Product in frequency domain ↔ convolution in time domain; useful for forcing functions

Watch Out

Convolution integral can be tedious; use only if partial fractions fail

When To Use

Inverse transforming products that do not factor into standard forms

Term

Region of Convergence (ROC)

Example

For f(t) = e^(at), ROC is s > a (ensures e^((a-s)t) → 0 as t → ∞)

Definition

Set of s values (typically s > σ for some σ) for which the Laplace integral converges; must be specified for unique F(s)

Term

Inverse Laplace Transform

Example

L⁻¹{2/(s²+4)} = sin(2t) because L{sin(2t)} = 2/(s²+4)

Definition

Operation L⁻¹{F(s)} that recovers f(t) from F(s); typically done via partial fractions and lookup table

Values

• dy/dx = g(x)h(y)
• RHS factors into x and y parts
• All separable equations (simplest approach)
• Separate variables: ∫dy/h(y) = ∫g(x)dx
• Implicit or explicit form y = f(x, C)

Property

Separable

Values

• dy/dx + P(x)y = Q(x)
• Linear in y and y' only; y appears to first power
• Standard form or after rearrangement
• Multiply by μ = e^(∫P dx), then integrate
• Explicit y = [∫Q·μ dx + C]/μ

Property

Linear

Values

• M dx + N dy = 0 with ∂M/∂y = ∂N/∂x
• Check exactness condition first
• When separable or linear fails, or given as exact form
• Find F such that ∂F/∂x = M and ∂F/∂y = N; set F = C
• Implicit form F(x, y) = C

Property

Exact

Values

• Δ = b² − 4ac > 0
• m₁ ≠ m₂, both real
• y = C₁e^(m₁x) + C₂e^(m₂x)
• No oscillation; exponential growth/decay depending on sign of m

Property

Distinct Real

Values

• Δ = b² − 4ac = 0
• m₁ = m₂ = m (multiplicity 2)
• y = (C₁ + C₂x)e^(mx)
• Polynomial × exponential; slower approach to equilibrium than distinct roots

Property

Repeated Real

Values

• Δ = b² − 4ac < 0
• m = α ± βi where α = −b/(2a), β = √(4ac−b²)/(2a)
• y = e^(αx)(C₁cos(βx) + C₂sin(βx))
• Damped oscillation (α < 0) or growing oscillation (α > 0); frequency β

Property

Complex Conjugate

Values

• yₚ = Axⁿ + Bxⁿ⁻¹ + ⋯ + Z
• yₚ = x(Axⁿ + Bxⁿ⁻¹ + ⋯ + Z) [if 0 is root of char. eqn]
• To avoid linear dependence with yₕ
• n + 1 coefficients

Property

Pₙ(x) = polynomial degree n

Values

• yₚ = Ae^(ax)
• yₚ = Axe^(ax) [if e^(ax) ∈ yₕ]
• Ensures yₚ not already in yₕ
• 1 coefficient (or 2 if multiplied by x)

Property

e^(ax)

Values

• yₚ = Asin(bx) + Bcos(bx)
• yₚ = x(Asin(bx) + Bcos(bx)) [if sin/cos in yₕ]
• Avoid duplication with yₕ terms
• 2 coefficients (or 4 if multiplied by x)

Property

sin(bx) or cos(bx)

Values

• yₚ = e^(ax)(A₀xⁿ + A₁xⁿ⁻¹ + ⋯ + Aₙ)
• yₚ = x·e^(ax)(A₀xⁿ + A₁xⁿ⁻¹ + ⋯ + Aₙ)
• Multiply entire yₚ by x if e^(ax) matches a yₕ term
• n + 1 coefficients (doubled if multiplied by x)

Property

e^(ax)·Pₙ(x)

Values

• dy/dt = ky
• k = growth rate (>0) or decay rate (<0)
• y(0) = y₀ (initial population)
• y(t) = y₀e^(kt)

Property

Exponential Growth/Decay

Values

• dN/dt = −(ln2/t₁/₂)N
• t₁/₂ = half-life time period
• N(0) = N₀ (initial atoms)
• N(t) = N₀(1/2)^(t/t₁/₂) or N₀e^(−kt) where k = ln2/t₁/₂

Property

Radioactive Decay (Half-Life)

Values

• dT/dt = −k(T − Tₛ)
• k = cooling coefficient; Tₛ = ambient temperature (constant)
• T(0) = T₀ (initial object temperature)
• T(t) = Tₛ + (T₀ − Tₛ)e^(−kt)

Property

Newton's Cooling

Values

• dQ/dt = C_in·F_in − (Q/V)·F_out
• C_in = inflow concentration; F_in, F_out = flow rates; V = tank volume
• Q(0) = Q₀ (initial solute amount)
• First-order linear; solve using integrating factor

Property

Tank Mixing (Constant Volume V)

Values

• ζ < 1
• Δ < 0 (complex roots α ± βi)
• Complex conjugate: m = −c/(2m) ± i√(4mk−c²)/(2m)
• Oscillates with decreasing amplitude; α = −c/(2m) < 0 (damping)
• Infinite (asymptotic approach)

Property

Underdamped

Values

• ζ = 1
• Δ = 0 (repeated real root)
• Repeated: m = −c/(2m) (multiplicity 2)
• Returns to equilibrium fastest without overshooting; no oscillation
• Shortest (optimal for many applications)

Property

Critically Damped

Values

• ζ > 1
• Δ > 0 (distinct real roots)
• Distinct: m₁, m₂ = [−c ± √(c²−4mk)]/(2m), both < 0
• No oscillation; slow exponential decay to equilibrium
• Longer than critical damping

Property

Overdamped