Homogeneous Differential Equation Non-Homogeneous Differential Equation

1. Homogeneous Differential Equation Non-Homogeneous Differential Equation Submitted to : M. Nauman Zubair Submitted by: Sir Atif Semester :2nd Roll # : 9021

2. Definitions: A differential equation is called a homogeneous differential equation if it can be written in the form M(x,y)dx + N(x,y)dy = 0 where M and N are of the same degree. First order ‘ordinary differential equation’ is homogeneous if it has the following form. dy/dx =F(y/x) It is simply an equation where both coefficients of the differentials dx and dy are homogeneous. A linear homogeneous differential equation having form: L(y) = 0 Where L is differential oprator. A linear nth order differential equation of the form an(x) d^n(y)/d^n(x)+an-1(x)d^n-1y/dx^n-1+…+a0(x)y=0 is said to be homogenous differential equation.

3. Standard Form: A linear homogeneous ordinary differential equation with constant coefficients has the general form of where are all constants. Homogeneous function: In mathematics, a homogeneous function is a function with multiplicative scaling behavior: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor.

4. Formal definition: • Homogeneous functions are functions where the sum of the powers of every term are the same. So the first function below is homogeneous of degree 3, the second and third are not homogeneous. • This method of recognizing a homogeneous function does not work all of the time, but is useful for many examples. A more formal definition states that if a function is homogeneous it can be written in the following form: • OR:

5. Linear functions: • Any linear function is homogeneous of degree 1, since by the definition of linearity for all and . Similarly, any multilinear function is homogeneous of degree n, since by the definition of multilinearity for all and . It follows that the nth Fréchet derivative of a function between two Banach spaces X and Y is homogeneous of degree n.

6. The History of Homogeneous Differential Equations, 1670–1950: • Differential equations have been a major branch of pure and applied mathematicssince their inauguration in the mid 17th century. While their history has beenwell studied, it remains a vital field of on-going investigation, with the emergenceof new connections with other parts of mathematics, fertile interplay with applied subjects, interesting reformulation of basic problems and theory invarious periods, new vistas in the 20th century, and so on. In this meeting we considered some of the principal parts of this story, from the launch with Newton and Leibniz up to around 1950. • Differential equations’ began with Leibniz, the Bernoulli brothers and others from the 1680s, not long after Newton’s ‘fluxional equations’ in the 1670s. Applications were made largely to geometry and mechanics; isoperimetrical problems were exercises in optimisation. • An algorithm for solving 2nd order linear homogeneous differential equation: Kovaciac in 1986 perform different results.

7. Most 18th-century developments consolidated the Leibnizian tradition, extending its multi-variate form, thus leading to partial differential equations. Generalisation of isoperimetrical problems led to the calculus of variations. New figures appeared, especially Euler, Daniel Bernoulli, Lagrange and Laplace. Development of the general theory of solutions included singular ones, functional solutions and those by infinite series. Many applications were made to mechanics, especially to astronomy and continuous media. In the 19th century: general theory was enriched by development of the understanding of general and particular solutions, and of existence theorems More types of equation and their solutions appeared; for example, Fourier analysis and special functions. Among new figures, Cauchy stands out. Applications were now made not only to classical mechanics but also to heat theory, optics, electricity and magnetism, especially with the impact of Maxwell. Later Poincar´e introduced recurrence theorems, initially in connection with the three-body problem.

8. In the 20th century: general theory was influenced by the arrival of set theory in mathematical analysis; with consequences for theorisation, including further topological aspects. New applications were made to quantum mathematics, dynamical systems and relativity theory.

9. Applications: An analog computer study of Vander Pol’s equation and its resonse to write Gaussian Noice. • In 1st Order Homogeneous Differential Equations The general form of the solution of the homogeneous differential equation can be applied to a large number of physical problems Barometric pressure variation with altitude:

10. ii)-Discharge of a capacitor: • Differential Equation Applications: • i)-Simple harmonic motion: • Ii)-Simple pendulum: • iii)-Azimuthal equation, hydrogen atom: • iv)-Velocity profile in fluid flow.

11. Charging a Capacitor : • An application of non-homogeneous differential equations • A first order non-homogeneous differential equation • has a solution of the form : For the process of charging a capacitor from zero charge with a battery, the equation is

12. Using the boundary condition Q=0 at t=0 and identifying the terms corresponding to the general solution, the solutions for the charge on the capacitor and the current are: In this example the constant B in the general solution had the value zero, but if the charge on the capacitor had not been initially zero, the general solution would still give an accurate description of the change of charge with time. The discharge of the capacitor is an example of application of the homogeneous differential equation

13. Capacitor Discharge: • An application of homogeneous differential equations • A first order homogeneous differential equation has a solution of the form : For the process of discharging a capacitor which is initially charged to the voltage of a battery, the equation is

14. Using the boundary condition and identifying the terms corresponding to the general solution, the solutions for the charge on the capacitor and the current are: Since the voltage on the capacitor during the discharge is strictly determined by the charge on the capacitor, it follows the same pattern. In the civil engineering: On the structural side of civil engineering, beam theory is based on a 4th order differential equation. Solutions/approximations of the solutions of this are the basis behind all structural engineering topics. Beam loading, stress, strain, and deflection are all related this way.

15. Physical Problem for Civil Engineering: A physical problem of finding how much time it would take a lake to have safe levels of pollutant. To find the time, the problem is modeled as an ordinary differential equation. Mass of pollutant = Mass of pollutant entering – Mass of pollutant leaving which also gives Rate of change of mass of pollutant = Rate of change of mass of pollutant entering – Rate of change of mass of pollutant leaving. In fluids : Anything that involves rates of flow can be modeled using diff EW, such as water treatment plants, and many environmental applications (typically grouped with civil topics). Physical Problem for Ordinary Differential Equations: A physical problem of finding how much time it would take a trunnion to cool down in a refrigerated chamber. To find the time, the problem would be modeled as a ordinary differential equation