Lecture-0

Differential Equations

• Definition: A differential equation is an equation in which the unknown quantity is a function (not just a number) that is related to its derivatives.

• Example: The equation ( y_0(x) = 2y(x) ) where ( y_0(x) := \frac{dy}{dx} ).

Modelling

• Definition: Modelling is the process of describing physical phenomena with mathematical idealizations, usually with equations that must be solved.

• Importance: It helps us make sense of complex systems by simplifying and abstracting them into more manageable forms.

Importance of Differential Equations in Modelling

• Physical processes are often described by laws that involve relationships between rates of change.

• The derivative of a function characterizes the rate of change of that function, making differential equations natural tools for understanding dynamic systems.

Reminder on Calculus

Elementary Functions

• Review properties and graphs of:

  • Powers of x: ( x^n )

  • Roots of x: ( x^{1/n} )

  • Exponential function: ( e^x )

  • Logarithmic functions: ( , \ln(x) )

  • Trigonometric functions: ( \sin(x), \cos(x), \tan(x) )

  • Inverse trigonometric functions: ( \arcsin(x), ... )

  • Hyperbolic functions: ( \sinh(x), \cosh(x) )

  • Composition of functions, like ( f(x) = x^2 ), ( g(x) = \sin(x) ): ( g(f(x)) = \sin(x^2) ), ( f(g(x)) = \sin^2(x) )

  • Operations on functions: addition, subtraction, multiplication, division (e.g., ( f(x) = x^2 + 2x + 1 )).

Derivative of a Function and Properties

• Definition: The derivative ( f'(x) ) is defined as the rate of change of the function, given by: [ f'(x) := \frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ]

• Properties:

  1. ( (f(x) + g(x))' = f'(x) + g'(x) ) (Linearity)

  2. ( (f(x)g(x))' = f'(x)g(x) + f(x)g'(x) ) (Leibniz/Product Rule)

  3. ( (f(g(x)))' = f'(g(x))g'(x) ) (Chain Rule)

Derivatives of Elementary Functions

• ( \frac{d}{dx} x^n = nx^{n-1} )

• ( \frac{d}{dx} e^x = e^x )

• ( \frac{d}{dx} \ln x = \frac{1}{x} )

• ( \frac{d}{dx} \sin(x) = \cos(x) )

• ( \frac{d}{dx} \cos(x) = -\sin(x) )

• ( \frac{d}{dx} \arcsin(x) = \frac{1}{\sqrt{1-x^2}} )

• ( \frac{d}{dx} \arccos(x) = -\frac{1}{\sqrt{1-x^2}} )

• ( \frac{d}{dx} \arctan(x) = \frac{1}{1+x^2} )

• ( \frac{d}{dx} \sinh(x) = \cosh(x) )

• ( \frac{d}{dx} \cosh(x) = \sinh(x) )

Integral of a Function and Properties

• 
  

  The integral of a derivative:[ \int f'(x)dx = f(x) + c ]

• Properties:

  • 
    

    Linearity:[ \int (f(x) + g(x))dx = \int f(x)dx + \int g(x)dx ]

  • 
    

    Integration by parts:[ \int f'(x)g(x)dx = f(x)g(x) - \int f(x)g'(x)dx ]

Examples of Integrals

• Example: ( \int \ln(x)dx = x\ln(x) - \int \frac{x}{1} dx )

• Further problems to try:

  1. ( \int \arctan(x)dx )

  2. ( \int x \cos(x)dx ) (Use integration by parts)