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CBSE Class 12 Commerce Mathematics Differential Equations

CBSE Class 12 Commerce Mathematics Differential Equations:- we provide complete details of CBSE Class 12 Commerce Mathematics Differential

CBSE Class 12 Commerce Mathematics Differential Equations

Equations in this article.

CBSE Class 12 Commerce Mathematics Differential Equations

we discussed how to differentiate a given function f with respect to an independent variable, i.e., how to find f ′(x) for a given function f at each x in its domain of definition. Further, in the chapter on Integral Calculus, we discussed how to find a function f whose derivative is the function g, which may
also be formulated as follows:

For a given function g, find a function f such that
dy/dx = g(x), where y = f (x) … (1)

An equation of the form (1) is known as a differential equation. A formal definition will be given later.

These equations arise in a variety of applications, may it be in Physics, Chemistry, Biology, Anthropology, Geology, Economics etc. Hence, an in depth study of differential equations has assumed prime importance in all modern scientific investigations. In this chapter, we will study some basic concepts related to differential equation, general and particular solutions of a differential equation, formation of differential equations, some methods to solve a first order – first degree differential equation and some applications of differential equations in different areas.

CBSE Class 12 Commerce Mathematics Differential Equations:-Basic Concepts

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We are already familiar with the equations of the type:

x2 – 3x + 3 = 0 … (1)
sin x + cos x = 0 … (2)
x + y = 7 … (3)

Let us consider the equation:
x dy/dx+ y= 0 … (4)

We see that equations (1), (2) and (3) involve independent and/or dependent variable (variables) only but equation (4) involves variables as well as derivative of the dependent variable y with respect to the independent variable x. Such an equation is called a differential equation.

In general, an equation involving derivative (derivatives) of the dependent variable with respect to independent variable (variables) is called a differential equation. A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation, e.g.,

2 d y dydx dx+ ⎛ ⎞ ⎜ ⎟ = 0 is an ordinary differential equation …. (5)

Of course, there are differential equations involving derivatives with respect to more than one independent variables, called partial differential equations but at this stage we shall confine ourselves to the study of ordinary differential equations only. Now onward, we will use the term ‘differential equation’ for ‘ordinary differential equation’.

CBSE Class 12 Commerce Mathematics Differential Equations:-Order of a differential equation

Order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation. Consider the following differential equations:
dy/dx = ex … (6)

CBSE Class 12 Commerce Mathematics Differential Equations:-General and Particular Solutions of a Differential Equation

In earlier Classes, we have solved the equations of the type:
x2 + 1 = 0 … (1)
sin2 x – cos x = 0 … (2)

Solution of equations (1) and (2) are numbers, real or complex, that will satisfy the given equation i.e., when that number is substituted for the unknown x in the given equation, L.H.S. becomes equal to the R.H.S..
Now consider the differential equation
22 d y y 0dx+ = … (3)

In contrast to the first two equations, the solution of this differential equation is a function φ that will satisfy it i.e., when the function φ is substituted for the unknown y (dependent variable) in the given differential equation, L.H.S. becomes equal to R.H.S..

The curve y = φ (x) is called the solution curve (integral curve) of the given differential equation. Consider the function given by
y = φ (x) = a sin (x + b), … (4)

where a, b ∈ R. When this function and its derivative are substituted in equation (3), L.H.S. = R.H.S.. So it is a solution of the differential equation (3).

Function φ consists of two arbitrary constants (parameters) a, b and it is called general solution of the given differential equation. Whereas function φ1 contains no arbitrary constants but only the particular values of the parameters a and b and hence is called a particular solution of the given differential equation. The solution which contains arbitrary constants is called the general solution (primitive) of the differential equation. The solution free from arbitrary constants i.e., the solution obtained from the general solution by giving particular values to the arbitrary constants is called a particular solution of the differential equation.

Formation of a Differential Equation whose General Solution is given We know that the equation
x2 + y2 + 2x – 4y + 4 = 0 … (1)
represents a circle having centre at (– 1, 2) and radius 1 unit.

CBSE Class 12 Commerce Mathematics Differential Equations:-Procedure to form a differential equation that will represent a givenfamily of curves

(a) If the given family F1 of curves depends on only one parameter then it is represented by an equation of the form
F1 (x, y, a) = 0 … (1)

For example, the family of parabolas y2 = ax can be represented by an equation of the form f (x, y, a) : y2 = ax.

Differentiating equation (1) with respect to x, we get an equation involving y′, y, x, and a, i.e.,
g (x, y, y′, a) = 0 … (2)
The required differential equation is then obtained by eliminating a from equations (1) and (2) as
F(x, y, y′) = 0 … (3)
(b) If the given family F2 of curves depends on the parameters a, b (say) then it is represented by an equation of the from
F2 (x, y, a, b) = 0 … (4)
Differentiating equation (4) with respect to x, we get an equation involving y′, x, y, a, b, i.e.,
g (x, y, y′, a, b) = 0 … (5)
But it is not possible to eliminate two parameters a and b from the two equations and so, we need a third equation. This equation is obtained by differentiating equation (5), with respect to x, to obtain a relation of the form
h (x, y, y′, y″, a, b) = 0 … (6)

The required differential equation is then obtained by eliminating a and b from equations (4), (5) and (6) as
F (x, y, y′, y″) = 0 … (7)

CBSE Class 12 Commerce Mathematics Differential Equations

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