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# Karnataka Class 12 Commerce Maths Differential Equations Complete Notes

## Karnataka Class 12 Commerce Maths Differential Equations Complete Notes

Karnataka Class 12 Commerce Maths Differential Equations : Here our team members provides you Karnataka Class 12 Commerce Maths Differential Equations Complete Notes in pdf format. Here we gave direct links for you easy to download Karnataka Class 12 Commerce Maths Differential Equations Complete Notes, Karnataka Class 12 Commerce Maths Differential Equations topics are Definition, order and degree, general and particular solutions of a differential equation.Formation of differential equation whose general solution is given.Solution of differential equations by method of separation of variables solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:

dy/dx + py = q, where p and q are functions of x or constants.

dx/dy + px = q, where p and q are functions of y or constants.

### Karnataka Class 12 Commerce Maths Differential Equations Complete Notes

Karnataka Class 12 Commerce Maths Differential Equations : A differential equation is a mathematical equationthat relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two.

A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form.

If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

### Karnataka Class 12 Commerce Maths Differential Equations Complete Notes

Karnataka Class 12 Commerce Maths Differential Equations : Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Commonly used distinctions include whether the equation is: Ordinary/Partial, Linear/Non-linear, and Homogeneous/Inhomogeneous. This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts.

#### Ordinary differential equations

An ordinary differential equation (ODE) is an equation containing a function of one independent variable and its derivatives. The term “ordinary” is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.

Linear differential equations, which have solutions that can be added and multiplied by coefficients, are well-defined and understood, and exact closed-form solutions are obtained. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by elementary functions in closed form: Instead, exact and analytic solutions of ODEs are in series or integral form. Graphical and numerical methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, analytic solutions.

#### Partial differential equations

A partial differential equation (PDE) is a differential equation that contains unknown multi variable functions and their partial derivatives. (This is in contrast to ordinary differential equations, which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved in closed form, or used to create a relevant computer model.

PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalization in stochastic partial differential equations.

#### Linear differential equations

A differential equation is linear if the unknown function and its derivatives have degree 1 (products of the unknown function and its derivatives are not allowed) and nonlinear otherwise. The characteristic property of linear equations is that their solutions form an caffeine subspace of an appropriate function space, which results in much more developed theory of linear differential equations.

Homogeneous linear differential equations are a subclass of linear differential equations for which the space of solutions is a linear subspace i.e. the sum of any set of solutions or multiples of solutions is also a solution. The coefficients of the unknown function and its derivatives in a linear differential equation are allowed to be (known) functions of the independent variable or variables; if these coefficients are constants then one speaks of a constant coefficient linear differential equation.

#### Non-linear differential equations

Non-linear differential equations are formed by the products of the unknown function and its derivatives are allowed and its degree is > 1.There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extend ability of solutions for nonlinear differential equations, and well-possession of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier–Stokes existence and smoothness). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.

#### Equation order

Differential equations are described by their order, determined by the term with the highest derivatives. An equation containing only first derivatives is a first-order differential equation, an equation containing the second derivative is a second-order differential equation, and so on. Differential equations that describe natural phenomena almost always have only first and second order derivatives in them, but there are some exceptions such as the thin film equation which is a fourth order partial differential equation.

### Karnataka Class 12 Commerce Maths Differential Equations Complete Notes

Karnataka Class 12 Commerce Maths Differential Equations : Here are my online notes for my differential equations course that I teach here at Lamar University.  Despite the fact that these are my “class notes”, they should be accessible to anyone wanting to learn how to solve differential equations or needing a refresher on differential equations.

I’ve tried to make these notes as self contained as possible and so all the information needed to read through them is either from a Calculus or Algebra class or contained in other sections of the notes. A couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

1. Because I wanted to make this a fairly complete set of notes for anyone wanting to learn differential equations I have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here.  You will need to find one of your fellow class mates to see if there is something in these notes that wasn’t covered in class.
2. In general I try to work problems in class that are different from my notes.  However, with Differential Equation many of the problems are difficult to make up on the spur of the moment and so in this class my class work will follow these notes fairly close as far as worked problems go.  With that being said I will, on occasion, work problems off the top of my head when I can to provide more examples than just those in my notes.  Also, I often don’t have time in class to work all of the problems in the notes and so you will find that some sections contain problems that weren’t worked in class due to time restrictions.
3. Sometimes questions in class will lead down paths that are not covered here.  I try to anticipate as many of the questions as possible in writing these up, but the reality is that I can’t anticipate all the questions.  Sometimes a very good question gets asked in class that leads to insights that I’ve not included here.  You should always talk to someone who was in class on the day you missed and compare these notes to their notes and see what the differences are.
4. This is somewhat related to the previous three items, but is important enough to merit its own item.  THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!!  Using these notes as a substitute for class is liable to get you in trouble. As already noted not everything in these notes is covered in class and often material or insights not in these notes is covered in class.

### Karnataka Class 12 Commerce Maths Differential Equations Complete Notes

Karnataka Class 12 Commerce Maths Differential Equations :  A differential equation contains one or more terms involving derivatives of one variable (the dependent variable, y) with respect to another variable (the independent variable, x).

### Karnataka Class 12 Commerce Maths Differential Equations Complete Notes

Karnataka Class 12 Commerce Maths Differential Equations :  A differential equation is an equation involving derivatives. The order of the equation is the highest derivative occurring in the equation.

Here are some examples

The first four of these are first order differential equations, the last is a second order equation.

The first two are called linear differential equations because they are linear in the variable y, the first has an “inhomogeneous term” that is independent of y on the right, the second is a homogeneous linear equation since all terms are linear in y.

The first three of these are “separable” differential equations, since they can be rewritten as dx f(x) = dy g(y) for appropriate f and g.

If you know only the derivative of a function, you do not have enough information to determine it completely. You can therefore seek either a solution to a differential equation, or a general solution (which usually has a constant for each order of the equation in it) or a solution subject to some additional condition or conditions.

You can find the general solution to any separable first order differential equation by integration, (or as it is sometimes referred to, by “quadrature”). All you need do is to integrate both sides of the equation dx f(x) = dy g(y). Thus you can apply the numerical techniques of the previous chapter to each of these directly and solve them numerically, if you cannot integrate them exactly.

The question we address here is:

Suppose we have a first order differential equation that is not separable, so we cannot reduce its solution to quadratures directly. Can we apply the numerical techniques previously for doing integrals to the task of solving these equations?

The answer is yes and we show how below. There is indeed a complication which we discuss next, but it can be overcome.

The implication of this fact is, that any system whose behavior can be modeled by a first order differential equation, or even by a set of linked first order equations, can be solved numerically to any desired accuracy by a modern computer very quickly. This makes possible real time control of such systems and is of great value in engineering.

### Karnataka Class 12 Commerce Maths Differential Equations Complete Notes

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