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Basic Mathematics-II




                    Notes          Self Assessment

                                   Fill in the blanks:
                                   1.  An equation is basically the mathematical manner to portray a ............................... among
                                       two variables.

                                                            dy
                                   2.  An equation of the form     Py   Q , in which P & Q are functions of x alone or constant
                                                            dx
                                       is called a ............................... equation of the first order.

                                   3.  To locate one quantity from the other you ............................... the first by some number,
                                       then add a different number to the outcome.
                                   4.  Linear differential equation is commonly known as ................................ .

                                   5.  On integrating a linear equation of the first order, we get ............................... as required
                                       general solution.
                                   6.  If we are concerning with  two or more equations,  it is  enviable to have a systematic
                                       technique of identifying if the system is consistent and to discover all ............................... .
                                   7.  In actual physical problems, quantities normally are associated ..............................., so this
                                       equation is very generally utilized.
                                   8.  The system is said to be ............................... if it contains a solution.

                                   11.2 Equations Reducible to the Linear (Bernoulli’s Equation)


                                                            dy        n
                                   (i)  An equation of the form     Py   Qy                             …..(1)
                                                            dx
                                       where P and Q are functions of x only or constants known as Bernoulli’s  equation. It can
                                       be made linear.
                                                                n
                                       Dividing both sides of (1) by y , we have
                                                 
                                        y   n dy    Py  1 n    Q                                       …..(2)
                                           dx
                                               –n
                                       Putting  y  = z

                                       So that  (1 n  )y  n dy    dz  .
                                                      dx  dx

                                       or  y  n dy    1  .  dz .
                                             dx  1 n dx
                                                  
                                       Equation (2) becomes
                                         1   dz
                                                 Pz   Q
                                        1 n dx
                                         
                                           dz
                                       or      1 n Pz      1 n   .Q                               …..(3)
                                           dx
                                       which is a linear differential equation with z as the dependent variable.
                                   If n > 1, then we have to add the solution y=0 to the solutions found by means of the technique
                                   illustrated above.



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