Chapter 9
                                     DIFFERENTIAL EQUATIONS
                                9.1 Overview
                                (i)      An equation involving derivative (derivatives) of the dependent variable with
                                         respect to independent variable (variables) is called a differential equation.
                                (ii)     A differential equation involving derivatives of the dependent variable with
                                         respect to only one independent variable is called an ordinary differential
                                         equation and a differential equation involving derivatives with respect to more
                                         than one independent variables is called a partial differential equation.
                                (iii)    Order of a differential equation is the order of the highest order derivative
                                         occurring in the differential equation.
                                (iv)     Degree of a differential equation is defined if it is a polynomial equation in its
                                         derivatives.
                                (v)      Degree (when defined) of a differential equation is the highest power (positive
                                         integer only) of the highest order derivative in it.
                                (vi)     A relation between involved variables, which satisfy the given differential
                                         equation is called its solution. The solution which contains as many arbitrary
                                         constants as the order of the differential equation is called the general solution
                                         and the solution free from arbitrary constants is called particular solution.
                                (vii)    To form a differential equation from a given function, we differentiate the
                                         function successively as many times as the number of arbitrary constants in the
                                         given function and then eliminate the arbitrary constants.
                                (viii)  The order of a differential equation representing a family of curves is same as
                                         the number of arbitrary constants present in the equation corresponding to the
                                         family of curves.
                                (ix)   €Variable separable method is used to solve such an equation in which variables
                                         can be separated completely, i.e., terms containingxshould remain withdxand
                                         terms containing y should remain with dy.
                             180    MATHEMATICS
                           (x)    A function F (x, y) is said to be a homogeneous function of degree n if
                                  F (λx, λy )= λn  F (x, y) for some non-zero constant λ.
                           (xi)   A differential equation which can be expressed in the form dy = F (x, y) or
                                                                                                   dx
                                   dx  = G (x, y), where F (x, y) and G (x, y) are homogeneous functions of degree
                                   dy
                                  zero, is called a homogeneous differential equation.
                           (xii)  To solve a homogeneous differential equation of the type dy  = F (x, y), we make
                                                                                             dx
                                  substitution y =vx and to solve a homogeneous differential equation of the type
                                   dx  = G (x, y), we make substitution x = vy.
                                   dy
                            (xiii) A differential equation of the form dy  + Py = Q, where P and Q are constants or
                                                                      dx
                                  functions ofx only is known as a first order linear differential equation. Solution
                                  of such a differential equation is given by y (I.F.) = ∫(Q × I.F.)dx + C, where
                                  I.F. (Integrating Factor) = e∫Pdx .
                           (xiv)  Another form of first order linear differential equation is dx  + P x = Q , where
                                                                                              dy     1     1
                                  P  and Q  are constants or functions of y only. Solution of such a differential
                                    1       1
                                  equation is given by x (I.F.) =   (Q ×I.F.)dy+ C, where I.F. =       Pdy.
                                                                  ∫    1                             e∫ 1
                           9.2 Solved Examples
                           Short Answer (S.A.)
                           Example 1 Find the differential equation of the family of curves y = Ae2x + B.e–2x.
                           Solution y = Ae2x + B.e–2x
                                                                                     DIFFERENTIAL EQUATIONS181
                                               dy                           d2y
                                                   = 2Ae2x – 2 B.e–2x and      2  = 4Ae2x + 4Be–2x
                                               dx                           dx
                                            d2y            d2y
                           Thus             dx2  = 4y i.e., dx2 – 4y = 0.
                           Example 2Find the general solution of the differential equation dy = y .
                                                                                              dx    x
                           Solution         dy = y          ⇒ dy = dx        ⇒∫dy= ∫dx
                                            dx    x              y      x          y       x
                                                            ⇒ logy = logx + logc ⇒ y = cx
                           Example 3 Given that dy = yex and x = 0, y = e. Find the value of y when x = 1.
                                                    dx
                           Solution dy = yex ⇒ ∫ dy    =    ∫exdx     ⇒      logy = ex + c
                                     dx             y
                           Substituting x = 0 and y = e,we get loge = e0+ c, i.e., c = 0 ( loge = 1)
                                                                                        
                           Therefore, log y = ex.
                                   Now,substituting x = 1 in the above, we get log y = e ⇒ y = ee.
                           Example 4Solve the differential equation dy  + y = x2.
                                                                       dx     x
                           Solution The equation is of the type dy +Py= Q,which is a linear differential
                                                                     dx
                           equation.
                                         1dx
                           Now  I.F. = ∫ x    =elogx = x.
                           Therefore, solution of the given differential equation is
                              182    MATHEMATICS
                                                              x4
                                   y.x = ∫ xx2 dx , i.e. yx =    +c
                                                               4
                                          x3   c
                            Hence y = 4 + x .
                            Example 5Find the differential equation of the family of lines through the origin.
                            Solution Let y = mx be the family of lines through origin. Therefore, dy = m
                                                                                                        dx
                            Eliminating m, we get y = dy . x  or x dy  – y = 0.
                                                          dx          dx
                            Example 6Find the differential equation of all non-horizontal lines in a plane.
                            Solution The general equation of all non-horizontal lines in a plane is
                                   ax + by = c, where a ≠ 0.
                            Therefore, a dx +b = 0.
                                           dy
                            Again, differentiating both sides w.r.t. y, we get
                               d2x          d2x
                             a dy2  = 0 ⇒ dy2 = 0.
                            Example 7 Find the equation of a curve whose tangent at any point on it, different
                            from origin, has slope y+ y .
                                                         x
                            Solution Given dy= y+ y             = y 1+1
                                                                          
                                              dx        x               x
                                     ⇒ dy=1+ 1dx
                                                    
                                          y        x
                            Integrating both sides, we get
                                   logy = x + logx + c ⇒        log y = x + c
                                                                      
                                                                     x