Every changing system in engineering — a cooling engine, a charging capacitor, a growing population — follows a First Order ODE. These equations describe the rate of change of one variable with respect to another.
When your phone battery discharges, the voltage drop follows an ODE. When a hot metal cools in air, the temperature curve is governed by Newton's law — which IS a first order ODE. When a RL circuit responds to a sudden voltage, the current buildup obeys a linear ODE.
- Electrical Engineering: RL/RC circuit analysis (Kirchhoff's current law = ODE)
- Mechanical Engineering: Newton's law of cooling, spring-damper systems
- Chemical Engineering: Reaction rate equations, diffusion models
- Civil Engineering: Groundwater seepage, beam deflection
An ordinary algebraic equation like 2x + 3 = 7 asks: "Find x."
A differential equation like dy/dx + 2y = 4x asks: "Find a FUNCTION y(x) whose derivative satisfies this relationship."
The Power of "Rate of Change"
Think of y as the temperature of a cup of coffee. dy/dx is how fast the temperature changes. An ODE tells you the law of change — solving it gives you the actual temperature at any time.
The curves above show: same ODE structure, different initial conditions → different solution curves. The family of all solution curves = General Solution.
Standard form: dy/dx + P(x)·y = Q(x)
The secret weapon: Integrating Factor (IF). Multiply both sides by IF to make the left side a perfect derivative.
Step-by-Step Procedure
dy/dx + P(x)y = Q(x). Identify P(x) and Q(x) clearly. If equation is in terms of x as function of y, write dx/dy + Px = Q instead.Standard form: dy/dx + P(x)·y = Q(x)·yⁿ
This is non-linear because of yⁿ. The trick is a substitution that converts it to a linear ODE.
Bernoulli → Banish the nonlinearity: sub v = y^(1−n). If n=1, equation separable. If n=0, already linear.
Special cases: n=2 → Logistic equation (population growth model!)
Form: M(x,y)dx + N(x,y)dy = 0
Geometric Intuition: An exact equation means there exists a potential function F(x,y) such that dF = 0, so F(x,y) = C is the solution — a family of level curves!
Integrating Factors for Non-Exact Equations
Newton's Law of Cooling
The rate of cooling is proportional to the temperature difference between the body and surroundings.
Natural Growth and Decay
Adding +C while computing the Integrating Factor: e^(∫P dx + C)
IF = e^∫P dx only. The constant cancels and is redundant at this stage.
Checking exactness as ∂M/∂x = ∂N/∂y instead of ∂M/∂y = ∂N/∂x
Test: ∂M/∂y = ∂N/∂x (M is with dx, differentiate w.r.t. the OTHER variable y)
In Bernoulli, forgetting to multiply (1−n) after substituting v
After sub, both P and Q get multiplied by (1−n): dv/dx + (1−n)Pv = (1−n)Q
First order ODEs describe systems with ONE storage element (one capacitor, one inductor). But real systems like LCR circuits, vibrating beams, and spring-mass-damper systems have TWO or more energy storage elements — they need second (or higher) order ODEs.
When you pluck a guitar string, the vibration follows a 2nd order ODE. When your phone vibrates, it's a damped oscillator described by a 2nd order ODE. Higher order = richer physics!
For: (aD² + bD + c)y = f(x)
Finding CF — Auxiliary Equation Method
Replace D with m: solve am² + bm + c = 0
When PI shortcuts fail (non-standard f(x)), use Variation of Parameters — the universal method.
|y₁' y₂'|
LCR Series Circuit
Simple Harmonic Motion (SHM)
L (Inductance) ↔ Mass m — resists change in current/velocity
R (Resistance) ↔ Damping — dissipates energy
1/C (Capacitance) ↔ Spring constant — stores/restores energy
E(t) (EMF) ↔ Applied force F(t)
For complex roots α±iβ, writing CF = e^(αx)[C₁cosh + C₂sinh] instead of cos/sin
Complex roots always give cos(βx) and sin(βx). cosh/sinh is for real unequal roots sometimes.
When f(D) of e^(ax) = 0, stopping and saying "PI doesn't exist"
Use the failure rule: multiply by x and use f'(a). If that also fails, multiply by x² and use f''(a)/2!
Forgetting to divide by coefficient of y'' before applying variation of parameters
Standard form MUST have coefficient of y'' = 1. Divide the entire equation before using the formula.
An ODE governs how a system changes in ONE variable (time or space). But heat flows through a 2D plate — temperature T depends on BOTH x, y, and t. Sound waves travel in 3D. These need Partial Differential Equations.
- Heat Equation: ∂u/∂t = α(∂²u/∂x²) — temperature distribution in a rod
- Wave Equation: ∂²u/∂t² = c²(∂²u/∂x²) — vibrating string, sound waves
- Laplace Equation: ∂²u/∂x² + ∂²u/∂y² = 0 — steady-state heat, electrostatics
- Diffusion Equation: same as heat — pollutant spread in air/water
Geometric view: A PDE solution z = f(x,y) is a SURFACE in 3D space. The PDE constrains how this surface can curve and tilt. The general solution is a family of surfaces.
Method 1: Eliminate Arbitrary Constants
Given a relation with n arbitrary constants → differentiate n times to generate n additional equations → eliminate all constants.
Method 2: Eliminate Arbitrary Functions
Given z = f(ax+by) or z = f(u) + g(v) — differentiate and eliminate the arbitrary functions f, g.
Eliminating 1 arbitrary function → 1st order PDE. Eliminating 2 arbitrary functions → 2nd order PDE. The number of arbitrary elements = order of resulting PDE.
The standard 1st order quasi-linear PDE: P(x,y,z)p + Q(x,y,z)q = R(x,y,z)
Geometric Insight: Pp + Qq = R defines a direction field on surface z(x,y). The characteristic curves weave through this field. Two independent integrals of the auxiliary system define the general solution.
If no direct integration works, try multipliers l, m, n such that: lP + mQ + nR = 0 AND l dx + m dy + n dz = 0 is an exact differential. Then integrate to get one integral directly.
Form: (aD^n + bD^(n−1)D' + ... + kD'^n)z = f(x,y) where D = ∂/∂x, D' = ∂/∂y
Method is analogous to higher-order ODEs! Replace D → m, D' → 1 (or 1, m) to get auxiliary equation.
In Lagrange's method, writing the general solution as u + v = 0 instead of F(u,v) = 0
General solution is ANY functional relationship: φ(u,v) = 0 or u = f(v). Do not restrict to u = v!
For PDE CF, writing C₁e^(m₁x) + C₂e^(m₂x) as in ODE case
PDE CF uses arbitrary FUNCTIONS: φ₁(y+m₁x) + φ₂(y+m₂x). Not constants — full functions!
Physics is full of fields: temperature at every point in a room, wind velocity at every point in the atmosphere, electric field around a charge. These fields need vector calculus to describe how they change, flow, and rotate.
- Gradient: Find the direction of steepest temperature rise → where heat flows fastest
- Divergence: Detect sources and sinks in fluid flow → does fluid compress or expand here?
- Curl: Detect rotation in velocity fields → vorticity in turbulence, magnetic fields
- Directional Derivative: Rate of change in any chosen direction
Intuition: Imagine you're on a hilly landscape. The gradient at your feet points in the direction you'd walk to climb FASTEST. Its magnitude tells you how steep that climb is.
Intuition: Place a tiny balloon in a fluid flow field F. If the fluid expands the balloon → div F > 0 (source). If it compresses → div F < 0 (sink). If no change → div F = 0 (solenoidal — no sources or sinks).
Intuition: Drop a tiny paddle wheel into a fluid flow F. If the wheel spins → curl F ≠ 0 (rotational). If it doesn't spin → curl F = 0 (irrotational). The direction of curl F is the axis of rotation; magnitude = speed of spin.
|∂/∂x ∂/∂y ∂/∂z|
|F₁ F₂ F₃ |
Expanded Curl Formula
Computing directional derivative without converting direction vector to unit vector first
ALWAYS normalize: u = a/|a|. Then D_u φ = ∇φ · u. The formula requires a UNIT vector.
Forgetting the negative sign in the j-component of curl (middle term)
In det expansion: i gets +, j gets −, k gets +. Middle component of curl always has a minus.
Confusing div (gives scalar) with curl (gives vector)
∇·F = SCALAR (just one number). ∇×F = VECTOR (three components). Never mix these up!
Vector differentiation told us how fields change locally. Vector integration tells us the cumulative effect — total work done by a force, total fluid flux through a surface, total charge enclosed in a volume.
- Line Integral: Work done by a force field along a path (W = ∫F·dr)
- Surface Integral (Flux): Total fluid flow through a surface per unit time
- Green's Theorem: Converts line integral ↔ double integral (2D version of Stokes)
- Stokes' Theorem: Converts surface integral ↔ line integral (circulation = curl flux)
- Divergence Theorem: Converts volume integral ↔ surface integral (total outflow)
Intuition: Walk along a curved path in a force field. At each tiny step dr, the work done = F·dr (dot product = force component along motion × step length). Total work = sum of all these tiny contributions = line integral.
Parametric Evaluation
Intuition: Hold a net in a river. The flux = total volume of water passing through the net per second. Net flow depends on: water speed, net area, AND the angle between flow and net (cos θ factor = dot product with normal).
Intuition: The total circulation (rotation) around a closed boundary = the sum of all the local rotations inside. Like adding up all the tiny paddle wheels inside a region — their total spin equals the big circulation around the edge.
Green = 2D version of Stokes. ∂N/∂x − ∂M/∂y is the "2D curl" (z-component of ∇×F where F = Mi + Nj).
Area by Green's: A = ∮_C x dy = −∮_C y dx = ½∮_C (x dy − y dx)
Intuition: The total circulation of a vector field around the boundary of a surface = the total curl (rotation) through that surface. Green's theorem is the 2D special case of Stokes' theorem!
Intuition: The total fluid flowing OUT through a closed surface = the total "creation" of fluid (divergence) inside the volume. Count all sources inside → matches total outward flow. No fluid created = closed surface = net zero flux (solenoidal).
Green's Theorem (2D): ∮C → ∬R — boundary to area
Stokes' Theorem (3D): ∮C → ∬S — boundary curve to surface (curl)
Divergence Theorem (3D): ∯S → ∭V — surface to volume (divergence)
All three say: "Integral over boundary = integral of derivative over region." It's the same idea at different dimensions!
Using Stokes' theorem without checking if F has continuous derivatives on the surface
Stokes requires F and its partial derivatives to be continuous everywhere on and inside S. Singularities invalidate it.
In Green's theorem, traversing C clockwise instead of counterclockwise
Standard Green's theorem: C traversed counterclockwise (positive orientation). Clockwise gives negative of the double integral.
Mixing up which theorem to use: Stokes for closed surfaces
CLOSED surface → Divergence theorem (volume inside). OPEN surface with boundary curve → Stokes' theorem.
Unit I — First Order ODE
Unit II — Higher Order ODE
Equal: (C₁+C₂x)e^(mx)
Complex: e^αx(C₁cosβx+C₂sinβx)
Fail: multiply by x, use f'(a)