Formula Sheet: Differential Equations

First-Order Differential Equations

General Form and Terminology

Standard Form:

\[a_n(x)\frac{d^ny}{dx^n} + a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}} + \cdots + a_1(x)\frac{dy}{dx} + a_0(x)y = g(x)\]

Variables and Terms:

• y = dependent variable (function of x)
• x = independent variable
• n = order of the differential equation (highest derivative present)
• g(x) = forcing function or non-homogeneous term
• When g(x) = 0, the equation is homogeneous
• When g(x) ≠ 0, the equation is non-homogeneous

Separable Equations

Form:

\[\frac{dy}{dx} = f(x)g(y)\]

Solution Method:

\[\frac{dy}{g(y)} = f(x)dx\] \[\int \frac{dy}{g(y)} = \int f(x)dx + C\]

Variables:

• C = constant of integration
• Solution obtained by separating variables and integrating both sides

Linear First-Order Equations

Standard Form:

\[\frac{dy}{dx} + P(x)y = Q(x)\]

Integrating Factor Method:

\[\mu(x) = e^{\int P(x)dx}\]

General Solution:

\[y = \frac{1}{\mu(x)}\left[\int \mu(x)Q(x)dx + C\right]\]

Alternative Form:

\[y \cdot e^{\int P(x)dx} = \int Q(x)e^{\int P(x)dx}dx + C\]

Variables:

• μ(x) = integrating factor
• P(x) = coefficient function of y
• Q(x) = non-homogeneous term
• C = constant of integration (determined from initial conditions)

Exact Equations

Form:

\[M(x,y)dx + N(x,y)dy = 0\]

Exactness Condition:

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

Solution Method: The solution F(x,y) = C is found where:

\[\frac{\partial F}{\partial x} = M(x,y)\] \[\frac{\partial F}{\partial y} = N(x,y)\]

General Solution:

\[F(x,y) = \int M(x,y)dx + g(y) = C\]

where g(y) is determined by ensuring ∂F/∂y = N(x,y)

Integrating Factor for Non-Exact Equations:

If \(\frac{1}{N}\left(\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}\right)\) is a function of x only:

\[\mu(x) = e^{\int \frac{1}{N}\left(\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}\right)dx}\]

If \(\frac{1}{M}\left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\right)\) is a function of y only:

\[\mu(y) = e^{\int \frac{1}{M}\left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\right)dy}\]

Bernoulli Equations

Form:

\[\frac{dy}{dx} + P(x)y = Q(x)y^n\]

Substitution:

\[v = y^{1-n}\]

Transformed Linear Equation:

\[\frac{dv}{dx} + (1-n)P(x)v = (1-n)Q(x)\]

Variables:

• n = exponent (n ≠ 0, n ≠ 1)
• v = new dependent variable
• After solving for v, back-substitute to find y

Second-Order Linear Differential Equations

General Form

Standard Form:

\[a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = f(x)\]

Variables:

• a, b, c = constants
• f(x) = forcing function
• When f(x) = 0, equation is homogeneous
• When f(x) ≠ 0, equation is non-homogeneous

Homogeneous Equations with Constant Coefficients

Characteristic Equation:

\[ar^2 + br + c = 0\]

Characteristic Roots:

\[r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Discriminant:

\[\Delta = b^2 - 4ac\]

Case 1: Two Distinct Real Roots (Δ > 0)

Roots: \(r_1\) and \(r_2\) (both real and distinct)

\[y = C_1e^{r_1x} + C_2e^{r_2x}\]

Case 2: Repeated Real Roots (Δ = 0)

Root: \(r_1 = r_2 = r\)

\[y = (C_1 + C_2x)e^{rx}\]

Case 3: Complex Conjugate Roots (Δ <>

Roots: \(r = \alpha \pm i\beta\)

\[y = e^{\alpha x}(C_1\cos(\beta x) + C_2\sin(\beta x))\]

Where:

\[\alpha = -\frac{b}{2a}\] \[\beta = \frac{\sqrt{4ac - b^2}}{2a}\]

Variables:

• C₁, C₂ = arbitrary constants (determined from initial conditions)
• r, r₁, r₂ = characteristic roots
• α = real part of complex roots
• β = imaginary part of complex roots
• i = imaginary unit (√-1)

Non-Homogeneous Equations

General Solution Structure:

\[y = y_h + y_p\]

Variables:

• y_h = homogeneous (complementary) solution
• y_p = particular solution

Method of Undetermined Coefficients

Applicable when: f(x) consists of polynomials, exponentials, sines, cosines, or their products

Form of Particular Solution Based on f(x):

• If f(x) = \(P_n(x)\) (polynomial of degree n), try: \(y_p = A_nx^n + A_{n-1}x^{n-1} + \cdots + A_1x + A_0\)
• If f(x) = \(ke^{ax}\), try: \(y_p = Ae^{ax}\)
• If f(x) = \(k\cos(\omega x)\) or \(k\sin(\omega x)\), try: \(y_p = A\cos(\omega x) + B\sin(\omega x)\)
• If f(x) = \(ke^{ax}\cos(\omega x)\) or \(ke^{ax}\sin(\omega x)\), try: \(y_p = e^{ax}(A\cos(\omega x) + B\sin(\omega x))\)

Modification Rule: If any term in the assumed \(y_p\) duplicates a term in \(y_h\), multiply \(y_p\) by x (or \(x^2\) if necessary) until no duplication exists.

Variation of Parameters

For equation:

\[y'' + p(x)y' + q(x)y = f(x)\]

If homogeneous solutions are \(y_1\) and \(y_2\), particular solution:

\[y_p = u_1(x)y_1 + u_2(x)y_2\]

Where:

\[u_1 = -\int \frac{y_2f(x)}{W}dx\] \[u_2 = \int \frac{y_1f(x)}{W}dx\]

Wronskian:

\[W = \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix} = y_1y_2' - y_2y_1'\]

Variables:

• W = Wronskian (must be non-zero for linearly independent solutions)
• y₁, y₂ = linearly independent solutions of the homogeneous equation
• u₁, u₂ = functions to be determined

Higher-Order Linear Differential Equations

General n-th Order Linear Equation

Standard Form:

\[a_n\frac{d^ny}{dx^n} + a_{n-1}\frac{d^{n-1}y}{dx^{n-1}} + \cdots + a_1\frac{dy}{dx} + a_0y = f(x)\]

Homogeneous Equations with Constant Coefficients

Characteristic Equation:

\[a_nr^n + a_{n-1}r^{n-1} + \cdots + a_1r + a_0 = 0\]

Solution Rules:

• For each real root r of multiplicity 1: contributes \(Ce^{rx}\)
• For each real root r of multiplicity m: contributes \((C_1 + C_2x + C_3x^2 + \cdots + C_mx^{m-1})e^{rx}\)
• For each pair of complex roots α ± iβ of multiplicity 1: contributes \(e^{\alpha x}(C_1\cos(\beta x) + C_2\sin(\beta x))\)
• For each pair of complex roots α ± iβ of multiplicity m: contributes \(e^{\alpha x}[(C_1 + C_2x + \cdots)cos(\beta x) + (D_1 + D_2x + \cdots)\sin(\beta x)]\)

Laplace Transform Method

Definition and Properties

Laplace Transform Definition:

\[\mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} e^{-st}f(t)dt\]

Inverse Laplace Transform:

\[f(t) = \mathcal{L}^{-1}\{F(s)\}\]

Variables:

• s = complex frequency variable
• t = time domain variable (t ≥ 0)
• F(s) = transform of f(t)

Common Laplace Transforms

• \(\mathcal{L}\{1\} = \frac{1}{s}\), s > 0
• \(\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}\), s > 0, n = positive integer
• \(\mathcal{L}\{e^{at}\} = \frac{1}{s-a}\), s > a
• \(\mathcal{L}\{\sin(at)\} = \frac{a}{s^2 + a^2}\), s > 0
• \(\mathcal{L}\{\cos(at)\} = \frac{s}{s^2 + a^2}\), s > 0
• \(\mathcal{L}\{\sinh(at)\} = \frac{a}{s^2 - a^2}\), s > |a|
• \(\mathcal{L}\{\cosh(at)\} = \frac{s}{s^2 - a^2}\), s > |a|
• \(\mathcal{L}\{e^{at}\sin(bt)\} = \frac{b}{(s-a)^2 + b^2}\), s > a
• \(\mathcal{L}\{e^{at}\cos(bt)\} = \frac{s-a}{(s-a)^2 + b^2}\), s > a
• \(\mathcal{L}\{t^ne^{at}\} = \frac{n!}{(s-a)^{n+1}}\), s > a

Laplace Transform Properties

Linearity:

\[\mathcal{L}\{af(t) + bg(t)\} = a\mathcal{L}\{f(t)\} + b\mathcal{L}\{g(t)\}\]

First Derivative:

\[\mathcal{L}\{f'(t)\} = sF(s) - f(0)\]

Second Derivative:

\[\mathcal{L}\{f''(t)\} = s^2F(s) - sf(0) - f'(0)\]

n-th Derivative:

\[\mathcal{L}\{f^{(n)}(t)\} = s^nF(s) - s^{n-1}f(0) - s^{n-2}f'(0) - \cdots - f^{(n-1)}(0)\]

Integration:

\[\mathcal{L}\left\{\int_0^t f(\tau)d\tau\right\} = \frac{F(s)}{s}\]

First Shifting Theorem (s-shift):

\[\mathcal{L}\{e^{at}f(t)\} = F(s-a)\]

Second Shifting Theorem (t-shift):

\[\mathcal{L}\{u(t-a)f(t-a)\} = e^{-as}F(s)\]

where u(t-a) is the unit step function

Multiplication by t:

\[\mathcal{L}\{tf(t)\} = -\frac{dF(s)}{ds}\]

Division by t:

\[\mathcal{L}\left\{\frac{f(t)}{t}\right\} = \int_s^{\infty}F(\sigma)d\sigma\]

Convolution Theorem:

\[\mathcal{L}\{f(t) * g(t)\} = F(s)G(s)\]

where the convolution is:

\[f(t) * g(t) = \int_0^t f(\tau)g(t-\tau)d\tau\]

Unit Step Function

Definition:

\[u(t-a) = \begin{cases} 0, & t < a="" \\="" 1,="" &="" t="" \geq="" a="" \end{cases}\]="">

Laplace Transform:

\[\mathcal{L}\{u(t-a)\} = \frac{e^{-as}}{s}\]

Dirac Delta Function

Definition:

\[\delta(t-a) = \begin{cases} 0, & t \neq a \\ \infty, & t = a \end{cases}\]

with the property:

\[\int_{-\infty}^{\infty}\delta(t-a)dt = 1\]

Laplace Transform:

\[\mathcal{L}\{\delta(t-a)\} = e^{-as}\]

Solving Differential Equations Using Laplace Transforms

General Procedure:

1. Take Laplace transform of both sides of the differential equation
2. Apply initial conditions
3. Solve the resulting algebraic equation for Y(s) = ℒ{y(t)}
4. Use partial fraction decomposition if necessary
5. Take inverse Laplace transform to find y(t)

For equation:

\[ay'' + by' + cy = f(t)\]

After taking Laplace transform:

\[a[s^2Y(s) - sy(0) - y'(0)] + b[sY(s) - y(0)] + cY(s) = F(s)\]

Solve for Y(s):

\[Y(s) = \frac{F(s) + asy(0) + ay'(0) + by(0)}{as^2 + bs + c}\]

Systems of Differential Equations

Linear Systems

General Form (2×2 system):

\[\frac{dx}{dt} = ax + by\] \[\frac{dy}{dt} = cx + dy\]

Matrix Form:

\[\frac{d\mathbf{X}}{dt} = \mathbf{A}\mathbf{X}\]

Where:

\[\mathbf{X} = \begin{bmatrix} x \\ y \end{bmatrix}, \quad \mathbf{A} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\]

Solution by Eigenvalues and Eigenvectors

Characteristic Equation:

\[\det(\mathbf{A} - \lambda\mathbf{I}) = 0\]

For 2×2 matrix:

\[(a-\lambda)(d-\lambda) - bc = 0\] \[\lambda^2 - (a+d)\lambda + (ad-bc) = 0\]

Eigenvalues:

\[\lambda = \frac{(a+d) \pm \sqrt{(a+d)^2 - 4(ad-bc)}}{2}\]

Case 1: Two Distinct Real Eigenvalues

General Solution:

\[\mathbf{X} = C_1\mathbf{v}_1e^{\lambda_1 t} + C_2\mathbf{v}_2e^{\lambda_2 t}\]

Case 2: Repeated Real Eigenvalue

General Solution:

\[\mathbf{X} = C_1\mathbf{v}e^{\lambda t} + C_2(\mathbf{v}t + \mathbf{w})e^{\lambda t}\]

Case 3: Complex Eigenvalues (α ± iβ)

General Solution:

\[\mathbf{X} = e^{\alpha t}[C_1(\mathbf{u}\cos(\beta t) - \mathbf{v}\sin(\beta t)) + C_2(\mathbf{u}\sin(\beta t) + \mathbf{v}\cos(\beta t))]\]

Variables:

• λ, λ₁, λ₂ = eigenvalues
• v, v₁, v₂ = eigenvectors
• C₁, C₂ = constants determined from initial conditions
• I = identity matrix

Solution by Laplace Transform

For system:

\[\frac{d\mathbf{X}}{dt} = \mathbf{A}\mathbf{X} + \mathbf{F}(t)\]

Taking Laplace transform:

\[s\mathbf{X}(s) - \mathbf{X}(0) = \mathbf{A}\mathbf{X}(s) + \mathbf{F}(s)\]

Solving for X(s):

\[\mathbf{X}(s) = (s\mathbf{I} - \mathbf{A})^{-1}[\mathbf{X}(0) + \mathbf{F}(s)]\]

Numerical Methods

Euler's Method

For initial value problem:

\[\frac{dy}{dx} = f(x,y), \quad y(x_0) = y_0\]

Iterative Formula:

\[y_{n+1} = y_n + hf(x_n, y_n)\] \[x_{n+1} = x_n + h\]

Variables:

• h = step size
• n = step number
• y_n = approximate value of y at x_n
• Truncation error is O(h²) per step, O(h) globally

Improved Euler Method (Heun's Method)

Predictor:

\[y_{n+1}^* = y_n + hf(x_n, y_n)\]

Corrector:

\[y_{n+1} = y_n + \frac{h}{2}[f(x_n, y_n) + f(x_{n+1}, y_{n+1}^*)]\]

Variables:

• y_n+1* = predicted value
• Truncation error is O(h³) per step, O(h²) globally

Runge-Kutta Methods

Second-Order Runge-Kutta (RK2)

\[k_1 = hf(x_n, y_n)\] \[k_2 = hf(x_n + h, y_n + k_1)\] \[y_{n+1} = y_n + \frac{1}{2}(k_1 + k_2)\]

Fourth-Order Runge-Kutta (RK4)

\[k_1 = hf(x_n, y_n)\] \[k_2 = hf\left(x_n + \frac{h}{2}, y_n + \frac{k_1}{2}\right)\] \[k_3 = hf\left(x_n + \frac{h}{2}, y_n + \frac{k_2}{2}\right)\] \[k_4 = hf(x_n + h, y_n + k_3)\] \[y_{n+1} = y_n + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)\]

Variables:

• k₁, k₂, k₃, k₄ = weighted slopes at different points
• RK4 has truncation error O(h⁵) per step, O(h⁴) globally
• Most commonly used method for balance of accuracy and computational efficiency

Series Solutions

Power Series Method

Assume solution of form:

\[y = \sum_{n=0}^{\infty}a_nx^n = a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots\]

Derivatives:

\[y' = \sum_{n=1}^{\infty}na_nx^{n-1}\] \[y'' = \sum_{n=2}^{\infty}n(n-1)a_nx^{n-2}\]

Procedure:

1. Substitute series and derivatives into differential equation
2. Combine series to common powers of x
3. Equate coefficients of like powers of x to zero
4. Solve recurrence relation for coefficients a_n

Frobenius Method

For equations with regular singular point at x = 0:

\[y = x^r\sum_{n=0}^{\infty}a_nx^n = \sum_{n=0}^{\infty}a_nx^{n+r}\]

Indicial Equation: Determines the exponent r by substituting the series into the differential equation and equating the coefficient of the lowest power of x to zero.

Variables:

• r = exponent (found from indicial equation)
• a_n = coefficients determined from recurrence relation
• Method applies when x = 0 is a regular singular point

Boundary Value Problems

General Form

Differential equation on interval [a, b]:

\[y'' + p(x)y' + q(x)y = f(x)\]

With boundary conditions:

• Dirichlet conditions: y(a) = α, y(b) = β
• Neumann conditions: y'(a) = α, y'(b) = β
• Mixed conditions: combination of Dirichlet and Neumann

Eigenvalue Problems

Sturm-Liouville Problem:

\[\frac{d}{dx}\left[p(x)\frac{dy}{dx}\right] + [q(x) + \lambda r(x)]y = 0\]

With boundary conditions at x = a and x = b

Variables:

• λ = eigenvalue parameter
• p(x), q(x), r(x) = known functions
• y(x) = eigenfunction corresponding to eigenvalue λ

Special Differential Equations

Cauchy-Euler Equation

Form:

\[ax^2y'' + bxy' + cy = 0\]

Substitution:

\[y = x^r\]

Characteristic Equation:

\[ar(r-1) + br + c = 0\] \[ar^2 + (b-a)r + c = 0\]

Case 1: Two Distinct Real Roots (r₁ ≠ r₂)

\[y = C_1x^{r_1} + C_2x^{r_2}\]

Case 2: Repeated Real Root (r₁ = r₂ = r)

\[y = (C_1 + C_2\ln x)x^r\]

Case 3: Complex Conjugate Roots (r = α ± iβ)

\[y = x^\alpha[C_1\cos(\beta\ln x) + C_2\sin(\beta\ln x)]\]

Bessel's Equation

Standard Form:

\[x^2y'' + xy' + (x^2 - n^2)y = 0\]

General Solution:

\[y = C_1J_n(x) + C_2Y_n(x)\]

Variables:

• J_n(x) = Bessel function of the first kind of order n
• Y_n(x) = Bessel function of the second kind of order n
• n = order (non-negative real number)

Legendre's Equation

Standard Form:

\[(1-x^2)y'' - 2xy' + n(n+1)y = 0\]

General Solution:

\[y = C_1P_n(x) + C_2Q_n(x)\]

Variables:

• P_n(x) = Legendre polynomial of order n
• Q_n(x) = Legendre function of the second kind
• n = non-negative integer for polynomial solutions

Applications

Mechanical Vibrations

Spring-Mass-Damper System:

\[m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = F(t)\]

Variables:

• m = mass (kg or slug)
• c = damping coefficient (N·s/m or lb·s/ft)
• k = spring constant (N/m or lb/ft)
• x(t) = displacement from equilibrium (m or ft)
• F(t) = external forcing function (N or lb)

Natural Frequency:

\[\omega_n = \sqrt{\frac{k}{m}}\]

Damping Ratio:

\[\zeta = \frac{c}{2\sqrt{km}} = \frac{c}{2m\omega_n}\]

Damped Natural Frequency:

\[\omega_d = \omega_n\sqrt{1-\zeta^2}\]

System Behavior:

• Underdamped: ζ < 1="">
• Critically damped: ζ = 1 (fastest return without oscillation)
• Overdamped: ζ > 1 (slow return without oscillation)

RLC Electrical Circuits

Series RLC Circuit:

\[L\frac{d^2q}{dt^2} + R\frac{dq}{dt} + \frac{1}{C}q = E(t)\]

Or in terms of current (i = dq/dt):

\[L\frac{di}{dt} + Ri + \frac{1}{C}\int i\,dt = E(t)\]

Differentiated form:

\[L\frac{d^2i}{dt^2} + R\frac{di}{dt} + \frac{1}{C}i = \frac{dE(t)}{dt}\]

Variables:

• L = inductance (H)
• R = resistance (Ω)
• C = capacitance (F)
• q(t) = charge (C)
• i(t) = current (A)
• E(t) = voltage source (V)

Natural Frequency:

\[\omega_n = \frac{1}{\sqrt{LC}}\]

Damping Coefficient:

\[\alpha = \frac{R}{2L}\]

Population Growth Models

Exponential Growth:

\[\frac{dP}{dt} = kP\] \[P(t) = P_0e^{kt}\]

Logistic Growth:

\[\frac{dP}{dt} = kP\left(1 - \frac{P}{M}\right)\] \[P(t) = \frac{M}{1 + Ae^{-kt}}\]

Where:

\[A = \frac{M - P_0}{P_0}\]

Variables:

• P(t) = population at time t
• P₀ = initial population
• k = growth rate constant
• M = carrying capacity (maximum sustainable population)
• A = constant determined from initial conditions

Heat Transfer

Newton's Law of Cooling:

\[\frac{dT}{dt} = -k(T - T_s)\]

Solution:

\[T(t) = T_s + (T_0 - T_s)e^{-kt}\]

Variables:

• T(t) = temperature of object at time t
• T_s = surrounding (ambient) temperature
• T₀ = initial temperature
• k = cooling constant (k > 0)

Mixing Problems

Tank with inflow and outflow:

\[\frac{dA}{dt} = \text{(rate in)} - \text{(rate out)}\] \[\frac{dA}{dt} = r_{in}c_{in} - r_{out}\frac{A(t)}{V(t)}\]

Variables:

• A(t) = amount of substance in tank at time t
• V(t) = volume of solution in tank at time t
• r_in = inflow rate (volume/time)
• r_out = outflow rate (volume/time)
• c_in = concentration of incoming solution (mass/volume)

If r_in = r_out, then V(t) is constant