Learn Differential Equations and Integration Formulas

Master Differential Equations and Integration: First and Second Order Formulas from Basics to Advanced for better understanding and problem-solving skill.

Category	Formula	Description
Applications	`\frac{dy}{dt} = ky \implies y = y_0 e^{kt}`	Exponential growth/decay
Applications	`\frac{d^2x}{dt^2} + \omega^2x = 0`	Simple harmonic motion
Applications	`\frac{dT}{dt} = -k(T - T_{\text{env}})`	Newton's law of cooling

1. Differential Equations Formulas

1.1 First-Order Differential Equations

Category	Formula	Description
Differential Equations	`\frac{dy}{dx} = f(x, y)``	General first-order differential equation
Differential Equations	`\frac{dy}{dx} = g(x)h(y) \implies \int \frac{1}{h(y)} \, dy = \int g(x) \, dx + C`	Separable differential equation
Differential Equations	`\frac{dy}{dx} + P(x)y = Q(x) \implies y \cdot e^{\int P(x)dx} = \int Q(x) e^{\int P(x)dx} \, dx + C`	Linear first-order differential equation
Differential Equations	`y'' + p(x)y' + q(x)y = 0`	Homogeneous second-order linear differential equation

Category	Formula	Description
First-Order Linear	`\( \frac{dy}{dx} + P(x)y = Q(x) \)`	General form of a first-order linear differential equation.
Solution to First-Order Linear	`\( y = e^{-\int P(x)dx} \left( \int Q(x)e^{\int P(x)dx}dx + C \right) \)`	Solution of a first-order linear differential equation.
Separable Equation	`\( \frac{dy}{dx} = g(x)h(y) \)`	General form of a separable differential equation.
Solution to Separable Equation	`\( \int \frac{1}{h(y)}dy = \int g(x)dx + C \)`	Solution of a separable differential equation.
Exact Equation	`\( M(x, y)dx + N(x, y)dy = 0 \)``	Solution exists if `\( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \)`, with `\( \psi(x, y) = C \)`.

1.2 Second-Order Differential Equations

Category	Formula	Description
Second-Order Homogeneous	`\( \frac{d^2y}{dx^2} + P(x)\frac{dy}{dx} + Q(x)y = 0 \)`	General form of a second-order homogeneous differential equation.
Bernoulli's Equation	`\( \frac{dy}{dx} + P(x)y = Q(x)y^n \)`	Nonlinear equation reducible to linear by substitution.
Clairaut's Equation	`\( y = x\frac{dy}{dx} + f\left(\frac{dy}{dx}\right) \)`	Special type of first-order differential equation.
Cauchy-Euler Equation	`\( x^2\frac{d^2y}{dx^2} + ax\frac{dy}{dx} + by = 0 \)`	Solved based on roots of the auxiliary equation.

2. Integration Formulas

2.1 General Formulas

Category	Formula	Description
Integration	`\int f(x) \, dx = F(x) + C`	Basic indefinite integral
Integration	`\int_a^b f(x) \, dx = F(b) - F(a)`	Definite integral
Integration	`\int u \, dv = uv - \int v \, du`	Integration by parts
Integration	`\int f(g(x))g'(x) \, dx = \int f(u) \, du`	Substitution rule
Integration	`\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1`	Power rule for integration
Integration	`\int e^x \, dx = e^x + C`	Exponential function integration
Integration	`\int \sin(x) \, dx = -\cos(x) + C`	Integration of sine
Integration	`\int \cos(x) \, dx = \sin(x) + C`	Integration of cosine
Integration	`\int \sec^2(x) \, dx = \tan(x) + C`	Integration of secant squared

2.2 Advance Formulas

Category	Formula	Description
Power Rule	`\( \int x^n dx = \frac{x^{n+1}}{n+1} + C \; (n \neq -1) \)`	Integration of power functions.
Exponential Rule	`\( \int e^x dx = e^x + C \)`	Integral of the exponential function.
Logarithmic Rule	`\( \int \frac{1}{x}dx = \ln|x| + C \)`	Integral of `\( \frac{1}{x} \)`.
Sine	`\( \int \sin(x)dx = -\cos(x) + C \)`	Integral of `\( \sin(x) \)`.
Cosine	`\( \int \cos(x)dx = \sin(x) + C \)`	Integral of `\( \cos(x) \)`.
Tangent	`\( \int \tan(x)dx = \ln|\sec(x)| + C \)`	Integral of `\( \tan(x) \)`.
Integration by Parts	`\( \int u dv = uv - \int v du \)`	Formula for integration by parts.
Definite Integral	`\( \int_a^b f(x)dx = F(b) - F(a) \)`	Fundamental theorem of calculus.
Trigonometric Substitution	For `\( \sqrt{a^2 - x^2} \)`: Substitute `\( x = a\sin(\theta) \)` For `\( \sqrt{a^2 + x^2} \)`: Substitute `\( x = a\tan(\theta) \)`	Method for integrals involving radicals.
Partial Fractions	`\( \int \frac{P(x)}{Q(x)}dx \)`	Break into simpler fractions for integration.
Gaussian Integral	`\( \int_{-\infty}^\infty e^{-x^2}dx = \sqrt{\pi} \)`	Integral of the Gaussian function.
Beta Function	`\( B(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1}dt \)`	Special function in calculus.
Gamma Function	`\( \Gamma(n) = \int_0^\infty x^{n-1}e^{-x}dx \)`	Extends factorial to continuous values.