Differential Equations

Differential Equations

1. Solutions types

   • Separation of variables
     • When the differential equation can be written in the form of where f is the function of y only and g is the function of x only. Taking an initial condition, rewrite this problem as and then integrate on both sides
   • Integrating factor
     • Technique is used when the differential equation is of the form where p and q are both the functions of x only. First-order differential equation is of the form . where and are both and the first derivative of . The higher- order differential equation is an equation that contains derivatives of an unknown function which can be either a partial or ordinary derivative. It can be represented in any order.

2. Notations types

   • Leinbniz
     • Leibniz's notation, the derivative of is expressed as . When we have an equation we can express the derivative as .
   • Newton
     • Newton's notation, the derivative of is expressed as f and the derivative of is expressed as y
   • Lagrange
     • Lagrange's notation, the derivative of F is expressed as F

3. Order

   • First Order
     • The order of highest derivative in case of first order differential equations is 1. A linear differential equation has order 1. In case of linear differential equations, the first derivative is the highest order derivative. and are either constants or functions of the independent variable only
   • differential Equations are classified on the basis of the order. Order of a differential equation is the order of the highest derivative (also known as differential coefficient) present in the equation.
     • Second Order
       • When the order of the highest derivative present is 2, then it is a second order differential equation.
   • Degree
     • The degree of the differential equation is represented by the power of the highest order derivative in the given differential equation. The differential equation must be a polynomial equation in derivatives for the degree to be defined.

4. Linearity

   • A Differential Equation is an equation with a function and one or more of its derivatives

5. A Differential Equation is an equation with a function and one or more of its derivatives

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