Differential Equation

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What are differential equations?

An equation containing an independent variable, dependent variable and differential coefficients of dependent variable with respect to independent variable are called differential equations.

What is the order of a differential equation?

The order of a differential equation is the order of the highest order derivative appearing in the equation. Note- The order of a differential equation is always a positive integer.                                                                                                                                                              Question 1: Determine the order of differential equation  

Solution 1: The order of highest order derivative is 2. So, it is a differential equation of order 2.

What is the degree of a differential equation?

The degree of a differential equation is the degree of the highest order derivative, when differential coefficients are made free from radicals and fractions.

Question 1: Determine the degree of differential equation y’+5y = 0                                                                                                      Solution 1: In this equation, the highest order derivative present id y’ and the highest power raised to it is 1. Therefore, the degree is 1.

Let us look at some more questions in order to understand better.

Question 2: Determine the degree of differential equation d3y / dx – 6 ( dy / dx)2 – 4y = 0                                                          Solution 2: In this equation the power of highest order derivative is 1. So, it is a differential equation of degree 1.

Question 3: Determine the degree of differential equation ( d2y / dx2 )3  + ( dy / dx)2 + sin( dy/dx ) +1  = 0                        Solution 3: We have, ( d2y / dx2 )3  + ( dy / dx)2 + sin( dy/dx ) +1  = 0                                                                                                                    Therefore, ( y”’)3  + (y”)2 + sin (y’) + 1 = 0                                                                                                                                                                          Since y’ is in sin (y’)  it is not a polynomial equation in derivative , hence, degree is not defined.

Question 4: Find the order and degree, if defined, of the following differential equation, dy/dx – cosx = 0                          Solution 4: The highest order derivative present in the differential equation is dy/dx , so its order is one. It is a polynomial equation in y’ and the highest power raised to dy/dx is one, so its degree is one.

What are linear and non- linear differential equations?

If a differential equation when expressed in the form of a polynomial involves the derivatives and dependent variable in the first power and there are no product of these, and also the coefficient of the various terms are either constants or functions of the independent variable, then it is said to be linear differential equation.Otherwise, it is a non-linear differential equation.

A differential equation will be non-linear differential equation if its degree is more than one and any of the differential coefficient has exponent more than one. If the exponents of the dependent variables is more than one and the products containing dependent variable and its differential coefficients are present then it will be a non-linear differential equation. For instance, the differential equation  (x+ y2) dx – 2xy dy=0 is a non- linear differential equation, because the exponent of dependent variable y is 2 and it involves the product of y and dy/dx.

Formation of differential equations

Question 1: Form the differential equation representing the family of curves y=mx, where m is arbitrary constant.     Solution 1: We have y = mx                                                            ……..(1)                                                                                                                       Differentiating both sides of equation (1) with respect to x , we get,  dy/dx = m                                                                                                     Substituting the value of m in equation (1), we get y = dy/dx . x  or x.dy/dx  –  y = 0   which is free from the parameter m and hence this is the required differential equation.

Question 2: Form the differential equation of the family of curves y = a sin (bx+c), a and c being parameters.                  Solution 2: We have,  y = a sin (bx+c)                                                    …..(1)                                                                                                                Since the given equation contains two arbitrary constants, we shall differentiate it two times and we shall get a differential equation of second order.                                                                                                                                                                                                                              Differentiating equation (1) with respect to x, we get,  dy/dx = a b cos (bx +c)              ……(2)                                                                                Differentiating equation (2) with respect to x, we get, d2y / dx = -a b2  sin (bx + c)                                                                                                d2y / dx = – b2 y         [ using (1)]                                                                                                                                                                                        d2y / dx + b2 y = 0                                                                                                                                                                                                                                               Hence, this is the required differential equation of the given family of curves. 

Question 3: Form the differential equation representing the family of curves y = A cos (x+B), A and B are parameters.   

Solution 3: We have,  y = A cos (x+B)                                                                    ……(1)                                                                                                Since the given equation contains two arbitrary constants, we shall differentiate it two times and we shall get a differential equation of second order.                                                                                                                                                                                                                              Differentiating equation (1) with respect to x, we get,  dy/dx = – A sin (x +B)                 ……(2)                                                                              Differentiating equation (2) with respect to x, we get, d2y / dx = – A cos (x + B)                                                                                                    d2y / dx = –  y         [ using (1)]                                                                                                                                                                                              d2y / dx +  y = 0                                                                                                                                                                                                                                                      Hence, this is the required  differential equation of the given family of curves.                                           

Solution of differential equations          

The solution of a differential equation is a relation between the variables involved which satisfies the differential equation. Such a relation and the derivatives obtained therefrom when substituted in the differential equation, makes left hand, and a right hand sides identically equal.

For example, y = ex  is a solution of the differential equation dy/dx = y.                                                                                                                Consider the differential equation   d2y / dx2  + y = 0                                         ……(1)                                                                                                and, consider  y = A cos x + B sin x  , where A and B are arbitrary constants           …..(2)                                                                                Differentiating equation (2) with respect to x, we get, dy / dx  = – A sin x  + B cos x                                                                                          Differentiating this with respect to x , we get,     d2y / dx = – A cos x – B sin x                                                                                                        Therefore,    d2y / dx = – y                                                                                                                                                                                                    d2y / dx2  + y = 0                                                                                                                                                                                                              This shows that  y = A cos x + B sin x  satisfies the differential equation (1) and hence it is a solution of equation (1).                                            It can be easily verified that y = 3 cos x + 2 sin x,  y = A cos x,  y = B sin x etc. , are also solutions of (1).                                                          We find that the solution y = 3 cos x + 2 sin x does not contain any arbitrary constant whereas solutions y = A cos x,  y = B sin x contain only one arbitrary constant. The solution y = A cos x + B sin x contains two arbitrary constants, so it is known as the general solution of (1) whereas all other solutions are particular solutions.

General solution – The solution which contains as many as arbitrary constants as the order of the differential equations is called the general solution of the differential equation. For example,  y = A cos x + B sin x is the general solution of the differential equation  d2y/dx2  + y = 0 . But,  y = A cos x is not the general solution as it contains one arbitrary constant.

Particular solution – Solutions obtained by giving particular values to the arbitrary constants in the general solution of a differential equation is called a particular solution. For example,  y = 3 cos x + 2 sin x is a particular solution of the differential equation (1).

Some important questions based on NCERT textbook are given below which you can practice on your own to score good marks in your exams.

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