Find the monic characteristic equation from a solution of the corresponding differential equation?

What is the general solution of a differential equation?

It is the solution of a differential equation without there being any restrictions on the variables (No boundary conditions are given). Presence of arbitrary constants indicates a general solution, the number of arbitrary constants depending on the order of the differential equation.

What is the global solution of an ordinary differential equation?

The global solution of an ordinary differential equation (ODE) is a solution of which there are no extensions; i.e. you can't add a solution to the global solution to make it more general, the global solution is as general as it gets.

What is monge's Method?

Monge's method, also known as the method of characteristics, is a mathematical technique used to solve certain types of partial differential equations. It involves transforming a partial differential equation into a system of ordinary differential equations by introducing characteristic curves. By solving these ordinary differential equations, one can find a solution to the original partial differential equation.

What is the local solution of an ordinary differential equation?

The local solution of an ordinary differential equation (ODE) is the solution you get at a specific point of the function involved in the differential equation. One can Taylor expand the function at this point, turning non-linear ODEs into linear ones, if needed, to find the behavior of the solution around that one specific point. Of course, a local solution tells you very little about the ODE's global solution, but sometimes you don't want to know that anyways.

What is a numerical solution of a partial differential equation?

Some partial differential equations do not have analytical solutions. These can only be solved numerically.

What is impulsive system in differential equation?

A differential equation have a solution. It is continuous in the given region, but the solution of the impulsive differential equations have piecewise continuous. The impulsive differential system have first order discontinuity. This type of problems have more applications in day today life. Impulses are arise more natural in evolution system.

What is oscillatory solution in differential equations?

It happens when the solution for the equation is periodic and contains oscillatory functions such as cos, sin and their combinations.

The solution to the differential equation dydxx2y3 , where y (3) 3 is?

y = 43x3+45&oline;&oline;&oline;&oline;&oline;&oline;&oline;&oline;&oline;&oline;√4

Why do you need initial condition to solve differential equation?

The solution to a differential equation requires integration. With any integration, there is a constant of integration. This constant can only be found by using additional conditions: initial or boundary.

Free download solution differential equation of eight edition?

y=c1e^x + c2e^-x

How do you solve a differential equation for x?

Another method to solve differential equation is taking y and dy terms on one side, and x and dy terms on other side, then integrating on both sides.This is a general solution. So if we want to particular solution we choose initial conditions.

Main points of Legendre differential equation?

The Legendre differential equation is the second-order ordinary differential equation(1)which can be rewritten(2)The above form is a special case of the so-called "associated Legendre differential equation" corresponding to the case . The Legendre differential equation has regular singular points at , 1, and .If the variable is replaced by , then the Legendre differential equation becomes(3)derived below for the associated () case.Since the Legendre differential equation is a second-order ordinary differential equation, it has two linearly independent solutions. A solution which is regular at finite points is called a Legendre function of the first kind, while a solution which is singular at is called a Legendre function of the second kind. If is an integer, the function of the first kind reduces to a polynomial known as theLegendre polynomial.The Legendre differential equation can be solved using the Frobenius method by making a series expansion with ,(4)(5)(6)Plugging in,(7)(8)(9)(10)(11)(12)(13)(14)so each term must vanish and(15)(16)(17)Therefore,(18)(19)(20)(21)(22)so the even solution is(23)Similarly, the odd solution is(24)If is an even integer, the series reduces to a polynomial of degree with only even powers of and the series diverges. If is an odd integer, the series reduces to a polynomial of degree with only odd powers of and the series diverges. The general solution for an integer is then given by the Legendre polynomials(25)(26)where is chosen so as to yield the normalization and is ahypergeometric function.The associated Legendre differential equation is(27)which can be written(28)(Abramowitz and Stegun 1972; Zwillinger 1997, p. 124). The solutions to this equation are called the associated Legendre polynomials (if is an integer), or associated Legendre functions of the first kind (if is not an integer). The complete solution is(29)where is a Legendre function of the second kind.The associated Legendre differential equation is often written in a form obtained by setting . Plugging the identities(30)(31)(32)(33)into (◇) then gives(34)(35)