Differential Equations - ODEs, PDEs, and Solution Methods

Table of Contents

1. First-Order ODEs
2. Second-Order Linear ODEs
3. Higher-Order Linear ODEs
4. Systems of ODEs
5. Series Solutions
6. Qualitative Analysis
7. Laplace Transform
8. Partial Differential Equations
9. Classification of PDEs

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First-Order ODEs

General Form

$\frac{dy}{dt} = f(t, y)$

Separable Equations

Form: dy/dt = f(t)g(y) or M(t)dt + N(y)dy = 0

Solution:

1. Separate: dy/g(y) = f(t)dt
2. Integrate both sides
3. Solve for y (implicit or explicit)

Example: dy/dt = y²t dy/y² = t dt → -1/y = t²/2 + C → y = -2/(t² + C)

Linear First-Order ODEs

Form: dy/dt + p(t)y = q(t)

Solution Method - Integrating Factor:

$μ(t) = e^{\int p(t)dt}$

Multiply both sides by μ: $μ(t)\frac{dy}{dt} + μ(t)p(t)y = μ(t)q(t)$

Left side is derivative of μ(t)y: $\frac{d}{dt}[μ(t)y] = μ(t)q(t)$

Integrate: $y = \frac{1}{μ(t)}\int μ(t)q(t)dt$

Example: dy/dt + 2ty = t μ(t) = e^(2∫t dt) = e^(t²) e^(t²)(dy/dt + 2ty) = e^(t²)t d/d Filter [e^(t²)y] = e^(t²)t y = e^(-t²)[∫e^(t²)t dt + C]

Exact Equations

Form: M(x,y)dx + N(x,y)dy = 0

Exact if: ∂M/∂y = ∂N/∂x

Solution:

1. Find F such that ∂F/∂x = M, ∂F/∂y = N
2. F(x,y) = C is solution

Method: Integrate M with respect to x, compare with N.

Example: (3x²+2xy)dx + (x²+1)dy = 0 Check: ∂M/∂y = 2x = ∂N/∂x ✓ F = ∫(3x²+2xy)dx = x³+x²y + h(y) Compare: x²+1 = ∂F/∂y = x²+h’(y) → h’(y) = 1 → h(y) = y F = x³+x²y + y = C

Bernoulli Equations

Form: dy/dt + p(t)y = q(t)y^n

Substitution: u = y^(1-n) Then: du/dt + (1-n)p(t)u = (1-n)q(t)

Homogeneous Equations

Form: dy/dt = F(y/t)

Substitution: u = y/t, then dy/dt = u + t(du/dt)

Riccati Equations

Form: dy/dt = p(t) + q(t)y + r(t)y²

Can often be reduced to linear or separable with proper substitution.

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Second-Order Linear ODEs

General Form

$a(t)\frac{d^2y}{dt^2} + b(t)\frac{dy}{dt} + c(t)y = f(t)$

Homogeneous: f(t) = 0

Constant Coefficient Case

$ay'' + by' + cy = 0$

Characteristic equation: ar² + br + c = 0

Roots: $r_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Fundamental Solutions

Case 1: Distinct Real Roots (r₁ ≠ r₂) $y = C_1e^{r_1t} + C_2e^{r_2t}$

Case 2: Repeated Real Root (r₁ = r₂ = r) $y = (C_1 + C_2t)e^{rt}$

Case 3: Complex Conjugate Roots (r = α ± βi) $y = e^{αt}(C_1\cos βt + C_2\sin βt)$ or $y = Ae^{αt}\cos(βt - φ)$

Non-Homogeneous: Undetermined Coefficients

Form: ay” + by’ + cy = g(t)

Method:

1. Solve homogeneous equation
2. Find particular solution y_p

Guess based on g(t):

• Polynomial of degree n: try polynomial of same degree
• e^(αt): try Ae^(αt)
• sin(βt) or cos(βt): try Asin(βt) + Bcos(βt)
• e^(αt) times polynomial: try e^(αt) times polynomial
• Products: try product of guesses

Adjustment: If guess is solution to homogeneous equation, multiply by t.

Example: y” - 4y’ + 4y = t Homogeneous: (r-2)² = 0 → y_h = (C₁ + C₂t)e^(2t) Guess: y_p = At + B y_p” - 4y_p’ + 4y_p = -4A + 4(At+B) = t 4A = 1, -4A + 4B = 0 A = 1/4, B = 1/4 y_p = t/4 + 1/4

Variation of Parameters

For: y” + p(t)y’ + q(t)y = g(t) (coefficients not necessarily constant)

Method: If y₁, y₂ are independent solutions to homogeneous equation:

$y_p(t) = -y_1\int\frac{y_2g(t)}{W(y_1,y_2)}dt + y_2\int\frac{y_1g(t)}{W(y_1,y_2)}dt$

where W(y₁,y₂) = y₁y₂’ - y₁’y₂ is Wronskian.

Reduction of Order

If one solution y₁ known, find second linearly independent solution.

Substitution: y₂ = v(t)y₁ y₂’ = v’y₁ + vy₁’ y₂” = v”y₁ + 2v’y₁’ + vy₁”

Substitute into ODE, set w = v’: $y_1w' + (2y_1' + p(t)y_1)w = 0$

Solve for w, integrate to get v, multiply by y₁.

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Higher-Order Linear ODEs

General Form

$a_n(t)\frac{d^ny}{dt^n} + a_{n-1}(t)\frac{d^{n-1}y}{dt^{n-1}} + \cdots + a_1(t)\frac{dy}{dt} + a_0(t)y = f(t)$

Constant Coefficient Case

Characteristic equation: a_nr^n + a_(n-1)r^(n-1) + … + a_0 = 0

General solution:

• Distinct real root r: add e^(rt)
• Real root r with multiplicity m: e^(rt), te^(rt), …, t^(m-1)e^(rt)
• Complex pair α ± βi with multiplicity m: e^(αt)(cos βt, sin βt, t times, …, t^(m-1) times)

Fundamental set: n linearly independent solutions.

Non-Homogeneous

General solution: y = y_h + y_p

y_h from homogeneous equation y_p using undetermined coefficients or variation of parameters.

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Systems of ODEs

General Form

$\frac{d\mathbf{x}}{dt} = \mathbf{A}\mathbf{x} + \mathbf{b}(t)$

where x is n-vector, A is n×n matrix, b is n-vector.

Homogeneous Systems

$\frac{d\mathbf{x}}{dt} = \mathbf{A}\mathbf{x}$

Solution: x(t) = e^(At)c, where c is constant vector.

Matrix Exponential

Definition: $e^{\mathbf{A}t} = \mathbf{I} + \mathbf{A}t + \frac{(\mathbf{A}t)^2}{2!} + \frac{(\mathbf{A}t)^3}{3!} + \cdots$

Properties:

• e^0 = I
• If AB = BA, then e^(A+B) = e^A e^B
• (e^A)⁻¹ = e^(-A)

Diagonalization Method

If A is diagonalizable (A = PDP⁻¹):

$e^{\mathbf{A}t} = \mathbf{P}e^{\mathbf{D}t}\mathbf{P}^{-1}$

where e^(Dt) is diagonal with e^(λᵢt) entries.

General solution: x(t) = Pe^(Dt) P⁻¹ c

Can also be written as: x(t) = c₁v₁e^(λ₁t) + c₂v₂e^(λ₂t) + … + cₙvₙe^(λₙt)

where vᵢ are eigenvectors.

Phase Portrait

Two-dimensional system visualization.

Types of equilibrium (at origin):

• Node (sinks/sources): Real eigenvalues, same sign (asymptotically stable if negative)
• Saddle: Real eigenvalues, opposite signs (unstable)
• Spiral: Complex eigenvalues with nonzero real part (stable if Re(λ) < 0)
• Center: Pure imaginary eigenvalues (stable, not asymptotically stable)

Stability: If all eigenvalues have negative real parts.

Non-Homogeneous Systems

$\frac{d\mathbf{x}}{dt} = \mathbf{A}\mathbf{x} + \mathbf{b}(t)$

Method 1: Undetermined coefficients (if b(t) is polynomial, exponential, etc.)

Method 2: Variation of parameters x_p = Φ(t)∫Φ⁻¹(t)b(t)dt where Φ(t) is fundamental matrix (columns are solutions).

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Series Solutions

Power Series Method

For y” + p(x)y’ + q(x)y = 0:

Try: y = ∑∞_{n=0} a_nx^n

Substitute, find recurrence relation for a_n.

Ordinary Points vs. Singular Points

Ordinary point: p(x), q(x) analytic at x₀

Regular singular point: (x-x₀)p(x), (x-x₀)²q(x) analytic

Irregular singular point: Otherwise

Frobenius Method

For regular singular point at x = 0:

Try: y = x^r ∑∞{n=0} a_nx^n = ∑∞{n=0} a_nx^(n+r)

Substitute into ODE, find indicial equation for r.

Cases:

• Distinct roots (not differing by integer): Two independent solutions
• Double root or differing by integer: Need reduction method

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Qualitative Analysis

Autonomous Systems

First-order: dy/dt = f(y) Equilibrium where f(y) = 0.

Phase line analysis:

• If f(y) > 0, y increasing
• If f(y) < 0, y decreasing

Stability:

• f’(y₀) < 0: stable equilibrium
• f’(y₀) > 0: unstable equilibrium

Systems of ODEs

Analyze equilibrium points where dx/dt = dy/dt = 0.

Linearization: Near equilibrium, system ~ dξ/dt = Jξ where J is Jacobian.

Jacobian: $\mathbf{J} = \begin{bmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \end{bmatrix}$

Eigenvalues of J determine local behavior.

Lyapunov Stability

Stable: Small perturbations stay bounded

Asymptotically stable: Small perturbations → 0 as t → ∞

Unstable: Some perturbations grow

Poincaré-Bendixson Theorem

For 2D autonomous systems in bounded region with no equilibrium: closed orbit (limit cycle) exists.

Bifurcations

Qualitative change in behavior as parameter changes.

Types: Saddle-node, transcritical, pitchfork, Hopf bifurcation.

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Laplace Transform

Definition

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

Domain: s real or complex.

Properties

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

Derivatives: $\mathcal{L}\{f'(t)\} = sF(s) - f(0)$ $\mathcal{L}\{f''(t)\} = s^2F(s) - sf(0) - f'(0)$

Integrals: $\mathcal{L}\left\{\int_0^t f(\tau)d\tau\right\} = \frac{F(s)}{s}$

Translation: $\mathcal{L}\{e^{at}f(t)\} = F(s-a)$ $\mathcal{L}\{f(t-a)u(t-a)\} = e^{-as}F(s)$

where u(t) is unit step.

Convolution: $\mathcal{L}\{f * g\} = F(s)G(s)$

where (f * g)(t) = ∫₀ᵗ f(τ)g(t-τ)dτ.

Common Transforms

• δ(t): 1 (Dirac delta)
• u(t): 1/s (step function)
• t^n: n!/s^(n+1)
• e^(at): 1/(s-a)
• sin at: a/(s²+a²)
• cos at: s/(s²+a²)
• t^n e^(at): n!/(s-a)^(n+1)

Solving ODEs

Method:

1. Take Laplace transform of both sides
2. Solve algebraic equation for F(s)
3. Use inverse transform to get f(t)

Example: y” + y = e^(-t), y(0) = 1, y’(0) = 0 s²F(s) - s + F(s) = 1/(s+1) F(s) = s/(s²+1) + 1/[(s²+1)(s+1)] Use partial fractions: 1/[(s²+1)(s+1)] = 1/2 · 1/(s+1) - 1/2 · (s-1)/(s²+1) Inverse: y(t) = cos t + 1/2 e^(-t) - 1/2 cos t + 1/2 sin t

Inverse Transform

Use tables, partial fractions, completing square for quadratics.

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Partial Differential Equations

Classification

Order: Highest derivative order

Linear: Coefficients depend only on variables

Homogeneous: No forcing term

Second-order linear PDE: $a\frac{\partial^2u}{\partial x^2} + b\frac{\partial^2u}{\partial x\partial y} + c\frac{\partial^2u}{\partial y^2} + \cdots = 0$

Discriminant: Δ = b² - 4ac

Types:

• Hyperbolic: Δ > 0 (wave equation)
• Parabolic: Δ = 0 (diffusion equation)
• Elliptic: Δ < 0 (Laplace equation)

Separation of Variables

Method: Assume u(x,t) = X(x)T(t), substitute, split into ODEs.

Example - Heat equation: u_t = k u_xx on [0,L] with u(0,t) = u(L,t) = 0 Assume u(x,t) = X(x)T(t): XT’ = kX”T T’/T = kX”/X = constant (say -λk) ODEs: X” + λX = 0, T’ = -λkT With BC: X(0) = X(L) = 0 Eigenvalues: λ_n = (nπ/L)² Eigenfunctions: X_n(x) = sin(nπx/L) T_n(t) = e^(-λ_n kt) u(x,t) = Σ A_n sin(nπx/L) e^(-λ_n kt) Initial condition determines A_n via Fourier series.

Method of Characteristics

For first-order PDEs.

Quasi-linear: a(x,y,u)u_x + b(x,y,u)u_y = c(x,y,u)

Characteristic curves: dx/dt = a, dy/dt = b, du/dt = c

Solve ODE system.

Fourier Method

Fourier series: Expand unknown in series of orthogonal functions (sines, cosines, eigenfunctions).

Fourier sine series: u(x) = Σ B_n sin(nπx/L)

Fourier cosine series: u(x) = A₀/2 + Σ A_n cos(nπx/L)

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Classification of PDEs

Wave Equation (Hyperbolic)

$\frac{\partial^2u}{\partial t^2} = c^2\frac{\partial^2u}{\partial x^2}$

D’Alembert’s solution: $u(x,t) = \frac{1}{2}[f(x-ct) + f(x+ct)] + \frac{1}{2c}\int_{x-ct}^{x+ct} g(\xi)d\xi$

Heat Equation (Parabolic)

$\frac{\partial u}{\partial t} = k\frac{\partial^2u}{\partial x^2}$

Fundamental solution: $\phi(x,t) = \frac{1}{\sqrt{4\pi kt}}e^{-x^2/(4kt)}$

General: Convolution of initial condition with fundamental solution.

Laplace’s Equation (Elliptic)

$\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0$

Mean value property: u is average of boundary values.

Maximum principle: Maximum/minimum on boundary.

Poisson’s Equation

$-\nabla^2u = f$

Inhomogeneous Laplace equation.

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Supplementary D.E. Techniques (from mathematics_GPT)

Summary Tables

First-Order ODEs

Form	Method
Separable	Separate, integrate, solve for y
Linear	Integrating Factor: μ(t) = e^(∫p(t)dt), multiply, integrate
Exact	M(x,y)dx + N(x,y)dy = 0, check ∂M/∂y = ∂N/∂x, find F, F(x,y) = C
Bernoulli	u = y^(1-n), du/dt + (1-n)p(t)u = (1-n)q(t)
Homogeneous	u = y/t, dy/dt = u + t(du/dt)
Riccati	p(t) + q(t)y + r(t)y², often reduced to linear or separable

Second-Order Linear ODEs

Form	Method
General	a(t)y” + b(t)y’ + c(t)y = f(t)
Homogeneous	ay” + by’ + cy = 0, characteristic equation ar² + br + c = 0
Non-Homogeneous	ay” + by’ + cy = g(t), undetermined coefficients: solve homogeneous, guess y_p, adjust if necessary
Variation of Parameters	y_p(t) = -y_1∫(y_2g(t)/W(y_1,y_2))dt + y_2∫(y_1g(t)/W(y_1,y_2))dt
Reduction of Order	y₂ = v(t)y₁, y₂’ = v’y₁ + vy₁’, y₂” = v”y₁ + 2v’y₁’ + vy₁”, y_1w’ + (2y_1’ + p(t)y_1)w = 0

Systems of ODEs

Form	Method
General	d\mathbf{x}/dt = \mathbf{A}\mathbf{x} + \mathbf{b}(t)
Homogeneous	d\mathbf{x}/dt = \mathbf{A}\mathbf{x}, x(t) = e^(At)c
Non-Homogeneous	d\mathbf{x}/dt = \mathbf{A}\mathbf{x} + \mathbf{b}(t), undetermined coefficients or variation of parameters

Series Solutions

Form	Method
Power Series	y” + p(x)y’ + q(x)y = 0, try y = ∑∞_{n=0} a_nx^n, substitute, find recurrence relation
Frobenius	y = x^r ∑∞_{n=0} a_nx^n, substitute, find indicial equation

Qualitative Analysis

System Type	Analysis
Autonomous (First-order)	dy/dt = f(y), equilibrium where f(y) = 0, phase line analysis, stability
Systems of ODEs	equilibrium points, linearization, Jacobian, eigenvalues
Lyapunov Stability	small perturbations, asymptotically stable, unstable
Poincaré-Bendixson	2D autonomous systems, bounded region, no equilibrium, closed orbit (limit cycle)
Bifurcations	qualitative changes as parameter changes, types: saddle-node, transcritical, pitchfork, Hopf

Solution Checklists

First-Order ODEs

• Separable: Check if dy/dt = f(t)g(y) or M(t)dt + N(y)dy = 0.
• Linear: Check if dy/dt + p(t)y = q(t).
• Exact: Check if M(x,y)dx + N(x,y)dy = 0.
• Bernoulli: Check if dy/dt + p(t)y = q(t)y^n.
• Homogeneous: Check if dy/dt = F(y/t).
• Riccati: Check if dy/dt = p(t) + q(t)y + r(t)y².

Second-Order Linear ODEs

• General: Check if a(t)y” + b(t)y’ + c(t)y = f(t).
• Homogeneous: Check if ay” + by’ + cy = 0.
• Non-Homogeneous: Check if ay” + by’ + cy = g(t).
• Variation of Parameters: Check if y_p(t) = -y_1∫(y_2g(t)/W(y_1,y_2))dt + y_2∫(y_1g(t)/W(y_1,y_2))dt.
• Reduction of Order: Check if y₂ = v(t)y₁, y₂’ = v’y₁ + vy₁’, y₂” = v”y₁ + 2v’y₁’ + vy₁”, y_1w’ + (2y_1’ + p(t)y_1)w = 0.

Systems of ODEs

• General: Check if d\mathbf{x}/dt = \mathbf{A}\mathbf{x} + \mathbf{b}(t).
• Homogeneous: Check if d\mathbf{x}/dt = \mathbf{A}\mathbf{x}.
• Non-Homogeneous: Check if d\mathbf{x}/dt = \mathbf{A}\mathbf{x} + \mathbf{b}(t).

Series Solutions

• Power Series: Check if y” + p(x)y’ + q(x)y = 0.
• Frobenius: Check if y = x^r ∑∞_{n=0} a_nx^n.

Unique Explanation Shortcuts

1. Integrating Factor (μ(t)):

   • For dy/dt + p(t)y = q(t), μ(t) = e^(∫p(t)dt).
   • Multiply both sides by μ(t) to make the left side a derivative.

2. Exact Equations:

   • M(x,y)dx + N(x,y)dy = 0 is exact if ∂M/∂y = ∂N/∂x.
   • Find F such that ∂F/∂x = M and ∂F/∂y = N.
   • F(x,y) = C is the solution.

3. Variation of Parameters:

   • For y” + p(t)y’ + q(t)y = g(t), if y₁, y₂ are independent solutions to homogeneous, then y_p(t) = -y_1∫(y_2g(t)/W(y_1,y_2))dt + y_2∫(y_1g(t)/W(y_1,y_2))dt.
   • W(y₁,y₂) is the Wronskian.

4. Reduction of Order:

   • If y₁ is a solution, find y₂ = v(t)y₁.
   • y₂’ = v’y₁ + vy₁’.
   • y₂” = v”y₁ + 2v’y₁’ + vy₁”.
   • Substitute into ODE, set w = v’.
   • Solve y_1w’ + (2y_1’ + p(t)y_1)w = 0 for w.

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