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Differential Equations Cheatsheet

A comprehensive cheat sheet covering essential concepts, formulas, and methods for solving differential equations, including first-order, second-order, and higher-order equations.

First-Order Differential Equations

Basic Forms and Definitions

A differential equation is an equation involving derivatives of a function.

A first-order differential equation involves only the first derivative.

General form: dy/dx = f(x, y)

An explicit solution is a function y = φ(x) that satisfies the differential equation.

A general solution contains arbitrary constants.

An implicit solution is a relation G(x, y) = 0 that defines a solution implicitly.

An initial value problem (IVP) consists of a differential equation and an initial condition y(x₀) = y₀.

Separable Equations

Form

dy/dx = f(x)g(y)

Solution

∫ dy/g(y) = ∫ f(x) dx

Example

dy/dx = x/y => ∫ y dy = ∫ x dx => y²/2 = x²/2 + C

Linear Equations

Form

dy/dx + P(x)y = Q(x)

Integrating Factor

μ(x) = e^(∫ P(x) dx)

Solution

y(x) = (1/μ(x)) ∫ μ(x)Q(x) dx

Example

dy/dx + y = x => μ(x) = e^x => y(x) = e^(-x) ∫ e^x x dx = x - 1 + Ce^(-x)

Exact Equations

Form

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

Test for Exactness

∂M/∂y = ∂N/∂x

Solution

∫ M(x, y) dx + ∫ [N(x, y) - ∂/∂y ∫ M(x, y) dx] dy = C

Example

(2x + y)dx + (x + 3y^2)dy = 0 is exact. Solution: x^2 + xy + y^3 = C

Homogeneous Equations

Form

dy/dx = f(x, y), where f(tx, ty) = f(x, y) for all t.

Substitution

v = y/x or y = vx, then dy/dx = v + x(dv/dx)

Example

dy/dx = (x^2 + y^2) / (xy). Let y = vx. Resulting separable equation can be solved.

Bernoulli Equations

Form

dy/dx + P(x)y = Q(x)y^n

Substitution

v = y^(1-n)

Transformed Equation

dv/dx + (1-n)P(x)v = (1-n)Q(x) (linear in v)

Second-Order Linear Homogeneous Equations

General Form

ay'' + by' + cy = 0, where a, b, and c are constants.

The characteristic equation is ar^2 + br + c = 0.

The roots r₁ and r₂ determine the form of the general solution.

Distinct Real Roots (r₁ ≠ r₂)

General Solution

y(x) = c₁e^(r₁x) + c₂e^(r₂x)

Example

For y'' - 3y' + 2y = 0, r₁ = 1, r₂ = 2. So, y(x) = c₁e^x + c₂e^(2x)

Repeated Real Roots (r₁ = r₂ = r)

General Solution

y(x) = c₁e^(rx) + c₂xe^(rx)

Example

For y'' - 4y' + 4y = 0, r = 2. So, y(x) = c₁e^(2x) + c₂xe^(2x)

Complex Conjugate Roots (r = α ± βi)

General Solution

y(x) = e^(αx)(c₁cos(βx) + c₂sin(βx))

Example

For y'' + 2y' + 5y = 0, r = -1 ± 2i. So, y(x) = e^(-x)(c₁cos(2x) + c₂sin(2x))

Initial Value Problems

Given y(x₀) = y₀ and y'(x₀) = y₁, solve for c₁ and c₂ using the initial conditions.

Substitute x₀ into the general solution and its derivative, then solve the resulting system of equations.

Second-Order Linear Non-Homogeneous Equations

General Form

ay'' + by' + cy = g(x), where a, b, and c are constants and g(x) ≠ 0.

The general solution is y(x) = y_c(x) + y_p(x), where y_c(x) is the complementary solution and y_p(x) is a particular solution.

y_c(x) is the general solution to the homogeneous equation ay'' + by' + cy = 0.

Method of Undetermined Coefficients

Applicable when

g(x) is a polynomial, exponential, sine, cosine, or a combination of these.

Procedure

Assume a form for y_p(x) based on g(x), with undetermined coefficients. Substitute into the differential equation to find the coefficients.

Example (Polynomial)

If g(x) = x^2, assume y_p(x) = Ax^2 + Bx + C

Example (Exponential)

If g(x) = e^(kx), assume y_p(x) = Ae^(kx)

Example (Sine/Cosine)

If g(x) = sin(kx), assume y_p(x) = Acos(kx) + Bsin(kx)

Variation of Parameters

Formula

y_p(x) = u₁(x)y₁(x) + u₂(x)y₂(x)

Where

u₁'(x) = -y₂(x)g(x) / W(y₁, y₂)
u₂'(x) = y₁(x)g(x) / W(y₁, y₂)

Wronskian

W(y₁, y₂) = y₁y₂' - y₂y₁'

General Solution

y(x) = c₁y₁(x) + c₂y₂(x) + y_p(x)

Laplace Transforms

Definition

The Laplace Transform of a function f(t) is defined as:

F(s) = L{f(t)} = ∫₀^∞ e^(-st) f(t) dt

Where s is a complex number and the integral converges.

Basic Laplace Transforms

L{1}

1/s, s > 0

L{t^n}

n! / s^(n+1), s > 0, n is a non-negative integer

L{e^(at)}

1 / (s - a), s > a

L{sin(at)}

a / (s^2 + a^2), s > 0

L{cos(at)}

s / (s^2 + a^2), s > 0

Properties of Laplace Transforms

Linearity

L{af(t) + bg(t)} = aL{f(t)} + bL{g(t)}

Derivative

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

Second Derivative

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

Translation in s

L{e^(at)f(t)} = F(s - a)

Translation in t

L{f(t - a)u(t - a)} = e^(-as)F(s), where u(t) is the Heaviside step function

Convolution

L{(f * g)(t)} = F(s)G(s)

Solving Differential Equations with Laplace Transforms

  1. Take the Laplace transform of both sides of the differential equation.
  1. Use initial conditions and properties of Laplace transforms to express the equation in terms of F(s).
  1. Solve for F(s).
  1. Take the inverse Laplace transform of F(s) to find f(t).
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