Differential Equations - Cheat Sheet
Your last-minute rapid-fire reference for all differential equation formulas, methods, and exam-critical facts. Cover classification, first-order methods, higher-order linear equations, and engineering applications in under 30 minutes.
Classification & Fundamentals
Formulas
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
Important facts
- A first-order ODE requires ONE initial condition; second-order requires TWO (e.g., y and y' at a point)
- Number of arbitrary constants in general solution = order of the DE
- Linear ODEs have superposition property: if y₁ and y₂ are solutions, so is C₁y₁ + C₂y₂
- Nonlinear ODEs generally have NO superposition property
- Always verify your solution by substituting back into the original DE
Key definitions
Ordinary Differential Equation (ODE)
Example
dy/dx + 2y = x² (one independent variable x)
Equation relating a function of one independent variable to its derivatives; opposed to partial DEs with multiple independent variables.
Linear ODE
Example
y'' + 3y' + 2y = 5 is linear; (y')² + y = 0 is nonlinear
The dependent variable y and all its derivatives appear to the first power only, with no products of y and its derivatives.
Homogeneous ODE
Example
y' = 2y is homogeneous; y' = 2y + 3 is non-homogeneous
Right-hand side equals zero; e.g., y'' + 5y' + 6y = 0 (no forcing term).
Particular Solution
Example
y = x² + 1 satisfying y(0) = 1
A specific solution satisfying given initial or boundary conditions; contains no arbitrary constants.
General Solution
Example
y = C₁e^(2x) + C₂e^(3x) for a second-order ODE
Contains arbitrary constants (number = order of DE); represents all possible solutions.
Diagrams to know
- ODE classification tree (Order, Linear/Nonlinear, Homogeneous/Non-homogeneous branches)
- Solution method decision flowchart based on ODE type
First-Order ODE Methods
Formulas
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
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
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
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
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
Important facts
- All linear first-order ODEs can be solved using the integrating factor method
- For separable equations, isolate dy/h(y) on one side and g(x)dx on the other BEFORE integrating
- Exact equations yield an implicit solution F(x,y) = C; do not always try to solve for y explicitly
- If ∂M/∂y ≠ ∂N/∂x, try multiplying by an integrating factor (often depends on x only or y only)
- Always apply initial conditions AFTER finding the general solution to determine the arbitrary constant
Key definitions
Integrating Factor (IF)
Example
For dy/dx + 2y = x, μ = e^(∫2dx) = e^(2x)
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
Exact Equation
Example
(2x + y)dx + (x + 2y)dy = 0 is exact since ∂M/∂y = 1 = ∂N/∂x
Differential equation M dx + N dy = 0 where ∂M/∂y = ∂N/∂x; solution is F(x,y) = C (implicit form)
Singular Solution
Example
For dy/dx = y², general solution is y = -1/(x+C); singular solution is y = 0
A solution not obtainable from the general solution for ANY value of the arbitrary constant; may arise from separable equations where h(y) = 0
Diagrams to know
- Method selection flowchart: Is it separable? → Linear? → Exact? → Needs IF?
- Integration factor construction diagram showing μ = e^(∫P dx)
Higher-Order Linear ODEs with Constant Coefficients
Formulas
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
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
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
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)
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
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
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
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
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)
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
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)
Important facts
- Solve characteristic equation FIRST; its roots determine the entire form of yₕ
- For distinct real roots: y = C₁e^(m₁x) + C₂e^(m₂x); for repeated root: y = (C₁ + C₂x)e^(mx); for complex α ± βi: y = e^(αx)(C₁cos βx + C₂sin βx)
- General solution of non-homogeneous ODE = yₕ + yₚ; ALWAYS find yₕ first
- If your guess for yₚ duplicates a term in yₕ, multiply the entire yₚ by x
- Wronskian determinant must be non-zero for solutions to be linearly independent
- Initial conditions (y(0) and y'(0)) applied to y = yₕ + yₚ to find C₁ and C₂
Key definitions
Characteristic Equation
Example
For y'' - 5y' + 6y = 0, characteristic equation is m² - 5m + 6 = 0 with roots m = 2, 3
Polynomial equation obtained by substituting y = e^(mx) into homogeneous linear constant-coefficient ODE; roots determine form of solution
Complementary Function (yₕ)
Example
For y'' + y' = x, yₕ = C₁ + C₂e^(-x) (solution to y'' + y' = 0)
General solution to the homogeneous part of a non-homogeneous ODE; contains all arbitrary constants
Particular Solution (yₚ)
Example
For y'' + y' = x, yₚ = x² - 2x is a particular solution
Any single solution to the non-homogeneous ODE; contains NO arbitrary constants
Wronskian (W)
Example
For y₁ = e^(2x) and y₂ = e^(3x), W = e^(2x)·3e^(3x) - e^(3x)·2e^(2x) = e^(5x) ≠ 0
Determinant W = y₁y₂' - y₂y₁' used to check linear independence and in variation of parameters; W ≠ 0 iff y₁ and y₂ are linearly independent
Linear Independence
Example
e^(2x) and e^(3x) are linearly independent; e^(2x) and 2e^(2x) are linearly dependent
Two solutions y₁ and y₂ are linearly independent if no constant k makes y₁ = ky₂; essential for a valid general solution
Diagrams to know
- Characteristic equation solution tree (real distinct, real repeated, complex conjugate roots)
- Solution form template based on root type
- Non-homogeneous solution composition diagram (yₕ + yₚ)
Engineering Applications
Formulas
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
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
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
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
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)
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
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)
Common values
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
Important facts
- Growth/decay always results in exponential form y = y₀e^(kt); identify whether growth (k > 0) or decay (k < 0)
- Find k from ANY known data point: if population is P₀ at t = 0 and 2P₀ at t = 5, then k = ln(2)/5 ≈ 0.139 per unit time
- Newton's cooling assumes T_s (ambient) is constant; if ambient changes, problem becomes more complex
- In tank mixing, output concentration = current solute amount / tank volume; as solute is removed, concentration decreases
- Spring-mass-damper: underdamped systems oscillate with decreasing amplitude; critical damping is the 'ideal' for rapid approach to equilibrium without overshoot
- Half-life / Doubling time: For decay, N(t) = N₀(1/2)^(t/t₁/₂); for growth, N(t) = N₀·2^(t/t_double)
Key definitions
Decay Constant (k)
Example
For Cobalt-60 with t₁/₂ = 5.27 years, k = 0.693/5.27 ≈ 0.131 per year
Proportionality constant in dy/dt = -ky for decay processes; related to half-life by t₁/₂ = ln(2)/k
Damping Ratio (ζ)
Example
For m = 1 kg, c = 2 N·s/m, k = 1 N/m: ζ = 2/(2√1) = 1 (critically damped)
Dimensionless parameter ζ = c/(2√(mk)) that classifies oscillatory behavior; ζ < 1 = underdamped (oscillation); ζ = 1 = critically damped; ζ > 1 = overdamped
Time Constant (τ)
Example
RC circuit: τ = RC; for R = 1 kΩ, C = 1 μF, τ = 1 ms
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
Diagrams to know
- Exponential decay curve (y = y₀e^(-kt)) showing half-life intervals
- Damping classification diagram (overdamped, critically damped, underdamped response curves)
- Tank mixing process flow diagram (input concentration/rate, tank volume, output concentration/rate)
Laplace Transform Method (Alternative for Linear ODEs)
Formulas
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
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
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
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)
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
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
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)}
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
Important facts
- Laplace method converts ODE with initial conditions into algebraic equation (linear); solve algebraically for Y(s), then invert
- Always write differential equation in standard form and identify initial conditions y(0), y'(0), etc. BEFORE applying transform
- Partial fraction decomposition is the key step in the inverse transform; break rational F(s) into sum of standard forms
- Convolution theorem extends method to non-standard forcing functions; less common in exam problems
- Laplace method excels for piecewise forcing (Heaviside functions) and delayed responses
Key definitions
Region of Convergence (ROC)
Example
For f(t) = e^(at), ROC is s > a (ensures e^((a-s)t) → 0 as t → ∞)
Set of s values (typically s > σ for some σ) for which the Laplace integral converges; must be specified for unique F(s)
Inverse Laplace Transform
Example
L⁻¹{2/(s²+4)} = sin(2t) because L{sin(2t)} = 2/(s²+4)
Operation L⁻¹{F(s)} that recovers f(t) from F(s); typically done via partial fractions and lookup table
Diagrams to know
- Laplace transform method flowchart (ODE with ICs → Transform → Algebraic equation → Solve → Inverse transform → Solution)
- Standard Laplace transform pair reference table
Must remember
- CLASSIFICATION FIRST: Always identify order, degree, and type (separable/linear/exact/etc.) before choosing solution method.
- SEPARABLE: Get all y on one side, all x on the other, THEN integrate both sides. Remember the constant of integration!
- LINEAR FIRST-ORDER: Use integrating factor μ = e^(∫P(x)dx); multiply entire equation by μ, then recognize LHS as d/dx[μy].
- CHARACTERISTIC EQUATION for constant-coefficient homogeneous ODEs: Substitute y = e^(mx), get am² + bm + c = 0; roots determine solution form.
- SOLUTION FORMS based on characteristic roots: (1) Distinct real m₁, m₂ → y = C₁e^(m₁x) + C₂e^(m₂x); (2) Repeated m → y = (C₁ + C₂x)e^(mx); (3) Complex α ± βi → y = e^(αx)(C₁cos βx + C₂sin βx).
- NON-HOMOGENEOUS = yₕ + yₚ: Complementary function yₕ (solution to homogeneous part) + Particular yₚ (any solution to full ODE).
- UNDETERMINED COEFFICIENTS: Guess yₚ based on form of f(x); if guess matches a term in yₕ, multiply entire guess by x.
- EXPONENTIAL GROWTH/DECAY: y = y₀e^(kt); find k from one data point; k > 0 = growth, k < 0 = decay; use ln(2) ≈ 0.693 for half-life.
- NEWTON'S COOLING: dT/dt = −k(T − Tₛ) ⟹ T(t) = Tₛ + (T₀ − Tₛ)e^(−kt); Tₛ is ambient (constant), k > 0 is cooling rate.
- ALWAYS VERIFY your solution by substituting back into the original ODE; apply initial conditions AFTER finding general solution to find arbitrary constants.
Last minute tips
- READ THE PROBLEM CAREFULLY: Identify initial conditions, type of ODE, and what is being asked (general solution vs. particular solution). Many students lose points by solving the wrong type.
- IF STUCK ON METHOD: Try separable first (easiest); if not separable, check for linear form dy/dx + P(x)y = Q(x); if neither, try exact or look for a substitution. Do NOT guess randomly.
- CHARACTERISTIC EQUATION IS YOUR BEST FRIEND for constant-coefficient ODEs: Roots determine 90% of the answer; practice factoring quadratics quickly.
- FOR yₚ GUESSING (undetermined coefficients): Write down yₕ first, then compare your yₚ guess to it. If there is ANY overlap, multiply yₚ by x. This is the #1 mistake.
- LAPLACE TRANSFORM IS A SHORTCUT for initial value problems: If the problem gives y(0) and y'(0), consider Laplace; partial fractions is the bottleneck—practice it.
Comparison tables
First-Order ODE Methods Comparison
| Method | Recognizable Form | When to Use | Key Step | Solution Type | |
|---|---|---|---|---|---|
| Separable | 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) |
| Linear | 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 |
| Exact | 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 |
Characteristic Equation Roots vs. Solution Form
| Root Type | Discriminant / Condition | Roots | General Solution yₕ | Behavior |
|---|---|---|---|---|
| Distinct Real | Δ = b² − 4ac > 0 | m₁ ≠ m₂, both real | y = C₁e^(m₁x) + C₂e^(m₂x) | No oscillation; exponential growth/decay depending on sign of m |
| Repeated Real | Δ = b² − 4ac = 0 | m₁ = m₂ = m (multiplicity 2) | y = (C₁ + C₂x)e^(mx) | Polynomial × exponential; slower approach to equilibrium than distinct roots |
| Complex Conjugate | 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 |
Non-Homogeneous ODE: Forcing Function vs. Particular Solution Guess
| Forcing f(x) | Guess for yₚ (if NOT in yₕ) | Guess for yₚ (if matches yₕ term) | Why Multiply by x | Number of Unknowns |
|---|---|---|---|---|
| Pₙ(x) = polynomial degree n | 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 |
| e^(ax) | yₚ = Ae^(ax) | yₚ = Axe^(ax) [if e^(ax) ∈ yₕ] | Ensures yₚ not already in yₕ | 1 coefficient (or 2 if multiplied by x) |
| sin(bx) or cos(bx) | 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) |
| e^(ax)·Pₙ(x) | 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) |
Application: Growth/Decay vs. Cooling vs. Mixing
| Application | Differential Equation | Key Constant | Initial Condition | Solution Form |
|---|---|---|---|---|
| Exponential Growth/Decay | dy/dt = ky | k = growth rate (>0) or decay rate (<0) | y(0) = y₀ (initial population) | y(t) = y₀e^(kt) |
| Radioactive Decay (Half-Life) | 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 |
| Newton's Cooling | 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) |
| Tank Mixing (Constant Volume V) | 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 |
Damping Classification (Spring-Mass-Damper System)
| Regime | Damping Ratio ζ | Discriminant Δ = c² − 4mk | Root Type | Response Behavior | Time to Equilibrium |
|---|---|---|---|---|---|
| Underdamped | 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) |
| Critically Damped | 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) |
| Overdamped | 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 |
Diagrams
First-Order ODE Solution Method Decision Tree
Decision flowchart for selecting the appropriate first-order ODE solution method. Follow the diamond decision nodes to identify whether the equation is separable, linear, exact, or requires substitution.
Differential Equations Master Map
Comprehensive mind map organizing all differential equation topics, methods, and applications. Shows hierarchical relationships between classification, solution methods, and real-world engineering applications.
Higher-Order Constant-Coefficient ODE Solver
Step-by-step flowchart for solving linear constant-coefficient ODEs. Branches on discriminant of characteristic equation and handles both homogeneous and non-homogeneous cases with method selection for particular solutions.
Damping System Response States
State diagram showing the progression of damped oscillation systems from initial displacement through various damping regimes (underdamped, critically damped, overdamped) to equilibrium. Illustrates response characteristics of each regime.
Applied Problem Recognition and ODE Setup
Comprehensive flowchart for recognizing and solving applied differential equation problems. Maps physical scenarios (growth, cooling, mixing, oscillation) to their corresponding differential equations and solution pathways.