2) Fourier Series of Simple Functions

Simple functions, such as square waves and triangular waves, can be represented as Fourier series.
The Fourier series of a function can reveal its frequency content.
Fourier series can be used to approximate functions with a high degree of accuracy.
Different forms of Fourier series can be used to represent functions with different periods.

Important Formulas to Remember in these topic

- Examples of Fourier series of simple functions in (0, 2π) and (–π, π):

$$f(x) = x$$

$$f(x) = \begin{cases} -1, &-\pi < x < 0 \\ 1, &0 < x < \pi \end{cases}$$

1) Introduction to Fourier Series

Fourier series represent periodic functions as a sum of sine and cosine functions.
The coefficients of a Fourier series can be found using Euler's formulas.
Fourier series are used in signal processing, communication systems, and other fields.
Fourier series converge to the function they represent, under certain conditions.

Important Formulas to Remember in these topic

- Definition of Fourier series:

$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \Big[ a_n\cos(nx) + b_n\sin(nx) \Big]$$

- Euler's formulae for determining Fourier coefficients:

$$a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) dx$$

$$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\cos(nx) dx$$

$$b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\sin(nx) dx$$

4) Applications of Laplace Transforms

The Laplace transform is a powerful tool for solving linear ordinary differential equations up to second order with initial conditions
The Laplace transform reduces a differential equation to an algebraic equation, which can be solved using standard techniques
The Laplace transform can also be used to solve integral equations, partial differential equations, and other mathematical problems
The Laplace transform is widely used in engineering, physics, and other fields to model and analyze time-dependent systems and signals
The Laplace transform is a versatile tool that can be combined with other mathematical techniques, such as the Fourier transform and the method of residues, to solve more complex problems.

3) Inverse Laplace Transform

The inverse Laplace transform is used to find the original function from its Laplace transform
The inverse Laplace transform has properties similar to those of the Laplace transform, such as the shifting theorem, scaling property, and multiplication and division by s
Partial fraction decomposition and the convolution theorem can be used to find the inverse Laplace transform of rational functions and the convolution of two functions, respectively
The Bromwich integral is a formula that can be used to compute the inverse Laplace transform directly from the Laplace transform, although it requires complex analysis
The inverse Laplace transform is an important tool in solving differential equations and other mathematical problems involving time-dependent functions

Important Formulas to Remember in these topic

- Inverse Laplace Transform:

$$\mathcal{L}^{-1}\{F(s)\} = \frac{1}{2\pi i}\lim_{T\to\infty}\int_{\gamma-iT}^{\gamma+iT}e^{st}F(s)ds,$$

where F(s) is a Laplace transform, gamma is a real number

- Shifting Theorem for Inverse Laplace Transform:

$$\mathcal{L}^{-1}\{e^{as}F(s)\} = u(t-a)\mathcal{L}^{-1}\{F(s)\},$$

where F(s) is a Laplace transform, and u(t-a) is the Heaviside step function defined by:

- Change of Scale Property for Inverse Laplace Transform:

$$\mathcal{L}^{-1}\{F(\alpha s)\} = \frac{1}{\alpha}\mathcal{L}^{-1}\{F(s)\},$$

where F(s) is a Laplace transform, and alpha is a positive constant.

- Multiplication and Division by s:

$$\mathcal{L}\{tf(t)\} = -F'(s), \quad \mathcal{L}\{f'(t)\} = sF(s) - f(0),$$

where f(t) is a function of t, and F(s) is its Laplace transform.

- Inverse Laplace Transform by Partial Fractions:

$$\mathcal{L}^{-1}\left\{\frac{P(s)}{Q(s)}\right\} = \sum_{i=1}^n\sum_{j=1}^{m_i} \frac{A_{i,j}}{(s-s_i)^j}e^{s_it},$$

where P(s) and Q(s) are polynomials in s, Q(s) has distinct roots s_1, s_2, \ldots, s_n, m_i is the multiplicity of s_i as a root of Q(s), and A_{i,j} are constants determined by partial fraction decomposition.

- Convolution Theorem:

$$\mathcal{L}\{f(t)*g(t)\} = F(s)G(s),$$

where f(t) and g(t) are functions of t, F(s) and G(s) are their Laplace transforms, and * denotes convolution. This theorem can be used to solve linear ordinary differential equations up to second order with initial conditions.

2) Properties of Laplace Transforms

The Laplace transform has several properties, such as the shifting theorem, scaling property, and multiplication and division by s, that can be used to simplify computation of the transform of more complex functions
The shifting theorem allows the Laplace transform of a function to be shifted in time by a fixed amount, while the scaling property allows the transform to be scaled by a constant factor
Partial fraction decomposition can be used to compute the Laplace transform of rational functions, which can then be inverted using the inverse Laplace transform
The convolution theorem states that the Laplace transform of the convolution of two functions is equal to the product of their Laplace transforms, which can be used to solve differential equations

Important Formulas to Remember in these topic

Shifting theorem:

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

Change of scale property:

$$L\{f(cx)\} = \frac{1}{c}F\left(\frac{s}{c}\right)$$

Multiplication and division by $s$:

$$L\{t f(t)\} = -\frac{d}{ds}F(s)$$

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

ILT by using partial fractions:

$$F(s) = \frac{P(s)}{Q(s)} = \frac{A_1}{s-a_1} + \frac{A_2}{s-a_2} + \dots + \frac{A_n}{s-a_n}$$

$$f(t) = \sum_{i=1}^{n} A_i e^{a_it}$$

Convolution theorem:

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

1) Laplace Transforms of Elementary Functions

The Laplace transform is a mathematical tool used to transform a function of time into a function of a complex variable s, which simplifies solving differential equations and other mathematical problems
The Laplace transform of elementary functions, such as polynomials, exponentials, trigonometric functions, and unit step functions, can be found using standard formulas and tables
The Laplace transform has linearity, first and second shifting, and scaling properties that allow the transform of more complex functions to be computed
The Laplace transform of derivatives and integrals can be derived from the definition of the transform, and can be used to solve differential equations with initial conditions
Improper integrals can be evaluated using the Laplace transform by taking the limit of the transform as the upper limit of integration approaches infinity

Important Formulas to Remember in these topic

Linearity property:

$$L\{af(x)+bg(x)\} = aL\{f(x)\}+bL\{g(x)\}$$

First shifting property:

$$L\{f(x-a)u(a)\} = e^{-as}F(s)$$

Change of scale property:

$$L\{f(cx)\} = \frac{1}{c}F\left(\frac{s}{c}\right)$$

Multiplication and division by $t$:

$$L\{tf(t)\} = -\frac{d}{ds}F(s)$$

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

Unit step function:

$$u(t-a) = \begin{cases} 0, & t < a \\ 1, & t \geq a \end{cases}$$

LT of unit step function:

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

Second shifting property:

$$L\{f(t-a)u(t-a)\} = e^{-as}L\{f(t)\}$$

Evaluation of improper integrals:

$$L\left\{\int_{0}^{\infty} f(t) dt\right\} = \int_{0}^{\infty} e^{-st}f(t) dt$$

5) Second order linear differential equations

Second order linear differential equations involve the second derivative of the unknown function and can be written in the form y'' + p(x)y' + q(x)y = r(x)
Homogeneous second order linear differential equations with constant coefficients have characteristic equations that can be used to find their solutions
Non-homogeneous second order linear differential equations can be solved using the method of undetermined coefficients or the method of variation of parameters
Particular integrals can be found for specific types of non-homogeneous functions, such as eax, sin ax, cos ax, ax^2 + bx + c (a, b, c are real numbers)
Second order linear differential equations are used to model many physical phenomena, such as oscillations, waves, and electromagnetic fields.

Important Formulas to Remember in these topic

Explanation and solution methods for second order differential equations with constant coefficients, including:

- Homogeneous equations, such as:

$$\frac{d^2y}{dx^2} + ay = 0 \Rightarrow y = c_1\cos(\sqrt{a}x) +$c_2\sin(\sqrt{a}x)$$

where a is a constant and c_1 and c_2 are arbitrary constants.

- Non-homogeneous equations, such as:

$$\frac{d^2y}{dx^2} + ay = f(x)$$

where f(x) is a known function. One solution can be found by the method of undetermined coefficients, which involves finding a particular solution based on the form of f(x). The general solution can then be found by adding the homogeneous solution to the particular solution.

- Complex roots, where the homogeneous equation has the form:

$$\frac{d^2y}{dx^2} + a_1\frac{dy}{dx} + a_2y = 0$$

and the roots of the characteristic equation are complex, such as:

$$r_1 = -\alpha + i\beta, \quad r_2 = -\alpha - i\beta$$

The general solution can be written as:

$$y = e^{-\alpha x}(c_1\cos(\beta x) + c_2\sin(\beta x))$$

where alpha and beta are constants, and c_1 and c_2 are arbitrary constants determined by initial or boundary conditions.

4) First order differential equations

First order differential equations involve only the first derivative of the unknown function
Variable separable equations involve separating the variables in the equation and integrating each side to obtain the solution
Homogeneous equations involve replacing the function with a new variable that simplifies the equation, and then solving it using separation of variables
Exact equations involve finding an integrating factor that makes the equation exact, and then integrating to obtain the solution
Bernoulli's equation involves transforming a nonlinear first order equation into a linear equation using a substitution, and then solving it using an integrating factor

Important Formulas to Remember in these topic

Explanation and solution methods for first order differential equations, including:

- Variable separable equations, such as:

$$\frac{dy}{dx} = f(x)g(y) \Rightarrow \int\frac{1}{g(y)}dy = \int f(x)dx + C$$

- Homogeneous equations, such as:

$$\frac{dy}{dx} = f\left(\frac{y}{x}\right) \Rightarrow y = vx \Rightarrow \frac{dv}{dx} = \frac{v-f(v)}{x}$$

- Exact equations, such as:

$$M(x,y)dx + N(x,y)dy = 0 \Rightarrow \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \Rightarrow \text{solution } F(x,y) = C$$

- Linear differential equation of the form dy/dx+Py=Q, such as:

$$\frac{dy}{dx} + P(x)y = Q(x) \Rightarrow y = e^{-\int P(x)dx}\left(\int Q(x)e^{\int P(x)dx}dx + C\right)$$

- Bernoulli's equation, such as:

$$\frac{dy}{dx} + P(x)y = Q(x)y^n \Rightarrow z = y^{1-n} \Rightarrow \frac{dz}{dx} + (1-n)P(x)z = (1-n)Q(x)$$

3) Formation of differential equations

Differential equations can be formed using various methods, including physical principles, geometry, and other mathematical models
The process of forming a differential equation from a given situation involves identifying the relevant variables, determining how they are related, and representing this relationship as an equation involving derivatives
In some cases, the boundary conditions or initial conditions must also be considered to obtain a unique solution
Differential equations can be classified according to their properties, such as linearity and order, which can provide insight into their solutions
The ability to form differential equations is an important skill for scientists and engineers who need to model complex systems

Important Formulas to Remember in these topic

There are various methods for creating differential equations, including:

- Direct integration, such as:

$$\frac{dy}{dx} = kx \Rightarrow y = \frac{kx^2}{2} + C$$

- Separation of variables, such as:

$$\frac{dy}{dx} = \frac{y}{x} \Rightarrow \frac{dy}{y} = \frac{dx}{x} \Rightarrow \ln|y| = \ln|x| + C$$

2) Order and degree of a differential equation

The order of a differential equation is the highest order derivative in the equation
The degree of a differential equation is the highest power of the highest order derivative in the equation
The order and degree of a differential equation can affect the complexity of its solution
Most commonly encountered differential equations are first or second order, but higher order equations are also possible
It is important to correctly identify the order and degree of a differential equation before attempting to solve it

Important Formulas to Remember in these topic

The order of a differential equation is the highest derivative that appears in the equation, such as:

$$\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0$$

The degree of a differential equation is the power to which the highest derivative is raised, after the equation has been written in standard form, such as:

$$\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 + y = 0$$

1) Definition of a differential equation

A differential equation is an equation that involves an unknown function and its derivatives
Differential equations are used to model real-world phenomena in many fields, including physics, engineering, economics, and biology
A differential equation is called an ordinary differential equation (ODE) if it involves only one independent variable, and a partial differential equation (PDE) if it involves multiple independent variables
Solutions to differential equations can be found using various techniques, including separation of variables, integrating factors, and Laplace transforms
Differential equations are an important topic in mathematics and have many practical applications

Important Formulas to Remember in these topic

A differential equation is an equation that relates a function and its derivatives to one another, such as:

$$\frac{dy}{dx} = f(x,y)$$

8) Mean and RMS values, Trapezoidal rule and Simpson’s 1/3 Rule for approximation integrals:

The mean value of a function over an interval is the integral of the function divided by the length of the interval.
The RMS value of a function over an interval is the square root of the integral of the square of the function divided by the length of the interval.
The trapezoidal rule approximates the area under a curve by dividing the area into trapezoids.
Simpson's 1/3 rule approximates the area under a curve by dividing the area into parabolic segments

Important Formulas to Remember in these topic

$$\text{Mean value } = \frac{1}{b-a}\int_a^b f(x)dx$$

$$\text{RMS value } = \sqrt{\frac{1}{b-a}\int_a^b [f(x)]^2 dx}$$

$$\text{Trapezoidal rule } = \frac{b-a}{2n}[f(x_0) + 2f(x_1) + 2f(x_2) + \dots + 2f(x_{n-1}) + f(x_n)]$$

$$\text{Simpson's 1/3 Rule } = \frac{b-a}{6}\left[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \dots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)\right]$$

7) Application of Integration to find areas under plane curves and volumes of Solids of revolution

Integration can be used to find the area under a curve, which can represent the distance traveled by an object or the work done by a force.
It can also be used to find the volume of a solid of revolution, which is formed by rotating a curve around an axis.

Important Formulas to Remember in these topic

$$\text{Area } = \int_a^b f(x)dx$$

$$\text{Volume } = \int_a^b \pi y^2 dx$$

$$\text{Volume } = \int_a^b \pi R^2 dx$$

6) Definite Integrals and properties, Definite Integral as the limit of a sum:

A definite integral is the area under a curve between two endpoints.
It can be calculated using the limit of a sum of rectangles that approximate the area under the curve.
The properties of definite integrals include linearity, additivity, and symmetry.

Important Formulas to Remember in these topic

$$\int_a^b f(x)dx = F(b) - F(a)$$

$$\int_a^b f(x)dx = -\int_b^a f(x)dx$$

$$\int_a^b [f(x) \pm g(x)]dx = \int_a^b f(x)dx \pm \int_a^b g(x)dx$$

$$\int_a^b kf(x)dx = k\int_a^b f(x)dx$$

$$\int_a^b f(x)dx + \int_b^c f(x)dx = \int_a^c f(x)dx$$

$$\int_a^b f(x)dx = \lim_{n\to\infty}\frac{b-a}{n}\sum_{i=1}^{n} f(x_i)$$

5) Integration by parts:

Integration by parts involves integrating the product of two functions.
The technique involves choosing one function to differentiate and the other to integrate.
The resulting integral involves an integral of the product of the two functions, which can often be simplified.

Important Formulas to Remember in these topic

$$\int u\frac{dv}{dx}dx = uv - \int v\frac{du}{dx}dx$$

4) Integration of reducible and irreducible quadratic factors:

A reducible quadratic factor can be factored into linear factors and then integrated using standard integration rules.
An irreducible quadratic factor cannot be factored and requires a special rule for integration.
For example, the integral of 1/(x^2 + a^2) is (1/a) arctan(x/a).

Important Formulas to Remember in these topic

$$\int \frac{Ax + B}{(ax^2 + bx + c)^n}dx$$

3) Integration by substitution

Integration by substitution involves substituting a new variable for the variable of integration.
This is useful when the integrand contains a function that can be simplified by substitution.
The technique involves choosing a new variable, computing its derivative, and substituting into the integral.
The resulting integral is often simpler to solve.

Important Formulas to Remember in these topic

$$\int f(g(x))g'(x)dx = \int f(u)du$$

2) Integration by decomposition of the integrand, integration of trigonometric, algebraic, exponential, logarithmic and Hyperbolic functions

Integration by decomposition involves breaking down a complicated function into simpler functions that can be integrated separately.
Trigonometric functions include sine, cosine, tangent, etc. and have their own integration rules.
Algebraic functions include polynomials and rational functions.
Exponential functions involve e^x or a^x, where a is a constant.
Logarithmic functions involve ln(x) or log(x), where x is a positive real number.
Hyperbolic functions include sinh, cosh, tanh, etc. and also have their own integration rules.

Important Formulas to Remember in these topic

$$\int f(x) dx = \int g(x)h(x) dx $$

$$\int \sin^m(x) \cos^n(x) dx $$

$$\int \frac{P(x)}{Q(x)} dx$$

1) Indefinite Integral – Standard forms

An indefinite integral is the anti-derivative of a function.
Standard forms of indefinite integrals include basic rules such as the power rule, the constant rule, and the sum rule.
The power rule states that the integral of x^n is (1/(n+1)) x^(n+1).
The constant rule states that the integral of a constant is the constant times the variable of integration.
The sum rule states that the integral of a sum of functions is the sum of the integrals of each individual function.

Important Formula to Remember in these topic

$$\int f(x) dx$$

Partial Differentiation - Partial derivatives up to second order - Euler's theorem

Partial Derivative Formulas

Partial Differentiation

$$\frac{\partial z}{\partial x} = \lim_{h \to 0} \frac{f(x + h, y) - f(x, y)}{h}$$

$$\frac{\partial z}{\partial y} = \lim_{k \to 0} \frac{f(x, y + k) - f(x, y)}{k}$$

Partial Derivatives up to Second Order

Second Order Partial Derivatives: $$\frac{\partial^2z}{\partial x^2} \quad \frac{\partial^2z}{\partial y^2} \quad \frac{\partial^2z}{\partial x \partial y} = \frac{\partial^2z}{\partial y \partial x}$$

Euler's Theorem

$$\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y}\cdot\frac{dy}{dx} = 0$$