Differential Equations

Differential Equations

Differential Equations serve as a fundamental branch of mathematics that explores the relationships between functions and their derivatives. Widely applied across numerous scientific and engineering domains, these equations provide a powerful framework for modelling dynamic systems and understanding their behaviour over time. By formulating and solving Differential Equations, researchers and practitioners can analyze physical phenomena, predict population dynamics, optimize engineering processes, and delve into the complexities of fluid dynamics. Mastery of Differential Equations is essential for individuals in mathematics, physics, engineering, and other scientific fields, enabling them to tackle real-world challenges with confidence and precision.

Order of a Differential Equation

The order of a Differential Equation refers to the highest derivative present in the equation. It is a crucial concept in understanding and classifying differential equations based on their complexity and the number of derivatives involved. The order of a differential equation determines the number of initial or boundary conditions required to find a unique solution and provides insights into the behaviour and nature of the equation's solutions. By analyzing the order of a differential equation, mathematicians and scientists can determine appropriate solution techniques and gain a deeper understanding of the dynamics and phenomena described by the equation.

The first, second and third equations involve the highest derivative of first, second and third order respectively. Therefore, the order of these equations is 1, 2 and 3 respectively.

Degree of a differential equation

To study thedegree of a differential equation, the key point is that the differential equation must be a polynomial equation in derivatives, i.e., y′, y″, y″′ etc. Consider the following differential equations:

We observe that the first equation is a polynomial equation in y″′, y″ and y′ with degree 1, the second equation is a polynomial equation in y′ (not a polynomial in y though) with a degree of 2. The degree of such differential equations can be defined. But the third equation is not a polynomial equation in y′ and the degree of such a differential equation can not be defined.

By the degree of a differential equation, when it is a polynomial equation in derivatives, we mean the highest power (positive integral index) of the highest order derivative involved in the given differential equation.

NOTE:Order and degree (if defined) of a differential equation are always positive integers.

Example:Find the order and degree, if defined, of each of the following differential equations:

Solution
(i) The highest order derivative present in the differential equation is , so its order is one. It is apolynomial equation iny′ and the highest power raised tois one, so its degree is one.
(ii) The highest order derivative present in the given differential equation is, so its order istwo. It is a polynomial equation inandand the highest power raised to is one, soits degree is one.

(iii) The highest order derivative present in the differential equation is y′′′ , so its order is three. The given differential equation is not a polynomial equation in its derivatives and so its degree is not defined.

Separable Equations

Separable equations are a fundamental concept in differential equations that can be solved by separating the variables and integrating each side separately, offering a powerful technique for finding solutions.

Separable differential equation:

Example:

RC circuits: Charging:

Discharging:

Linear Differential Equations

, integrating factor:

Example: . , .

Exact Differential Equations

• Potential function: For , we can find a such that and ; is the potential function; is exact.
• Exact differential equation: a potential function exists; general solution: .

Example: .

• Theorem: Test for exactness:

Example: , .

Integrating Factors

• Integrating factor: such that is exact.

Example: .

• How to find integrating factor:

Example:

• Separable equations and integrating factors:
• Linear equations and integrating factors:

Homogeneous and Bernoulli Equations

• Homogeneous differential equation: ; let separable.

Example: .

• Bernoulli equation: ; linear; separable; otherwise, let linear

Example: .

Higher Order Linear Differential Equation

A higher order linear differential equation is a mathematical equation that relates a function to its derivatives of various orders, where the highest order derivative is linearly related to the function itself.

Homogeneous Linear ODEs

A linearordinary differential equationof order is said to be homogeneous if it isof the form

where

, i.e., if all the terms are proportional to a derivative of (or itself) and there is no term that contains a function of alone.

However, there is also another entirely different meaning for afirst-order ordinary differential equation. Such an equation is said to be homogeneous if it can be written in the form

1. a nth order ODE if the nth derivative of the unknown function is the highest occurring derivative.
2. Linear ODE: .
3. Homogeneous linear ODE: .

Theorem: Fundamental Theorem for the Homogeneous Linear ODE: For a homogeneous linear ODE, sums and constant multiples of solutions on some open interval I are again solutions on I. (This does not hold for a Non-Homogeneous or nonlinear ODE!).

General solution: , where is a basis (or fundamental system) of solutions on I; that is, these solutions are linearly independent on I.

1. Linear independence and dependence: n functions are called linearly independent on some interval I where they are defined if the equation on I implies that all are zero. These functions are called linearly dependent on I if this equation also holds on I for somenot all zero.

Example: . Sol.: .

Theorem: Let the homogeneous linear ODE have continuous coefficients, on an open interval I. Then n solutions on I are linearly dependent on I if and only if their Wronskian/Determinant is zero for some in I. Furthermore, if W is zero for, then W is identically zero on I. Hence if there is an in I at which W is not zero, thenare linearly independent on I, so that they form a basis of solutions of the homogeneous linear ODE on I.

Wronskian:

1. Initial value problem: An ODE with n initial conditions , , .

Homogeneous Linear ODEs with Constant Coefficients

Homogeneous linear ordinary differential equations (ODEs) with constant coefficients are a class of ODEs where the derivatives of an unknown function are linearly related, and the coefficients of the derivatives are constant values throughout the equation.

1. : Substituting, we obtain the characteristic equation .

(i) Distinct real roots: The general solution is

Example: . Sol. .

(ii) Simple complex roots: , ,.

Example: . Sol..

(iii) Multiple real roots: Ifis a real root of order m, then m corresponding linearly independent solutions are: , , , .

Example: . Sol..

(iv) Multiple complex roots: If are complex double roots, the corresponding linearly independent solutions are: , , , .

1. Convert the higher-order differential equation to a system of first-order equations.

Example: .

Nonhomogeneous Linear ODEs

Nonhomogeneous linear ordinary differential equations (ODEs) are mathematical equations that involve derivatives of an unknown function, where the equation is not equal to zero. These equations play a vital role in various fields of science and engineering.

1. , the general solution is of the form: , where is the homogeneous solution and is a particular solution.
2. Method of undermined coefficients

Example: . Sol. .

1. Method of variation of parameters: , where , .

Example: .

Sol. .