Tiêu đề DIFFERENTIAL EQUATIONS (Q).pdf ✅

\\ Chapter – 1 Generally, any equation, such as f (x, y, a) = 0 .... (i) represents for each individual value of a, a member of a family of curves. Sometimes it is found necessary to represent the whole family of curves as a single unit and consider them as one for the purpose of studying a common property or characteristic which may run through the members of the family. From the given equation, solve for a, and the equation  (x, y ) = a may be obtained; and on differentiating, ‘a’ gets removed. The resulting equation involving dx dy is known as a differential equation i.e. the equation representing all the members of the family f (x, y, a ) = 0 or alternately  (x, y )  a. An equation involving an independent variable x, a dependent variable y and the differential coefficients of the dependent variable i.e. 2 2 , dx d y dx dy , ......etc is known as a differential equation. It can also be expressed as a function of variables x, y and derivatives of y w.r.t. x such as , ,   0      dx dy f x y Geometrically, differential equations represent a family of curves having a common property. To form a differential equation, we differentiate the given family of curves and eliminate the unknown constants as follows: (i) Consider the equation y = ax. This represents the Cartesian equation to a family of straight lines through the origin. Differentiating y = ax, we get a. dx dy  Eliminating a, we get the differential equation x dx dy y   . Hence dx dy y  x is the differential equation of all straight lines passing through the origin. (ii) Consider another example, the equation . 2 2 2 x  y  a This, for various a, represents a family of concentric circles with centre at origin. INTRODUCTION 1 THEORY CONTENT OF DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATION 2 FORMATION OF DIFFERENTIAL EQUATION 3
Differentiating the relation we get 2  2  0 dx dy x y (a is eliminated) i.e. x + y  0 dx dy which may be said to be the differential equation to a family of concentric circles. (iii) Now consider another equation representing a family of curves in the form f (x, y, a, b) = 0 .... (i) containing two arbitrary constants. In this case, since there are two constants, it becomes necessary to differentiate equation twice so that the result contains dx dy and 2 2 dx d y and can be expressed in the form , , , 0 2 2          dx d y dx dy F x y .... (ii) This equation is said to represent a differential equation of the family of curves represented by equation (i). Thus in the case of y = ax + b a dx dy  0 2 2  dx d y which is the differential equation of the family of all straight lines. Illustration 1 Question: Form the differential equation of the following relation: (i) x 2 + y 2 = 2ax (ii) x 2 + y 2 = 2ax + b (iii) y = aex + be2x Solution: (i) Consider the relation x 2 + y 2 = 2ax Differentiating, a dx dy 2x  2y  2 Eliminating a,          dx dy x y x 2x 2y 2 2  2 0 2 2    dx dy x y xy In this case the relation contains only one constant and hence the differential equation contains only dx dy . (ii) Consider the relation x 2 + y 2 = 2ax + b Differentiating a dx dy 2x  2y  2 Differentiating once again, 1 0 2 2 2          dx d y y dx dy which is the differential equation to the given equation and since there are two constants a and b, the differential equation contains (the second order) derivative 2 2 dx d y . (iii) Consider the relation y = aex + be2x x x ae be dx dy 2   2 x x ae be dx d y 2 2 2   4
Consider 3 2 4 3( 2 ) 2 ( ) 2 2 2 2 2 x x x x x x y ae be ae be ae be dx dy dx d y          For the relation y = aex + be2x , we get the (second order) differential equation 3 2 0 2 2   y  dx dy dx d y . As we know that an equation containing an independent variable, a dependent variable and the derivatives of the dependent variable, is called a differential equation. It has an order and degree defined as follows: 4.1 ORDER OF A DIFFERENTIAL EQUATION The highest derivative occurring in a differential equation defines its order. 4.2 DEGREE OF A DIFFERENTIAL EQUATION The power of the highest order derivative occurring in a differential equation is called the degree of the differential equation, for this purpose the differential equation is made free from radicals and fractions of derivatives. 4.3 EXAMPLES Differential equation Order of D.E. Degree of D.E.  y x dx dy  4  sin 1 1  x y e dx dy dx d y                  4 5 2 2 2 4  y x dx dy dx d y 3 cos 2 2    2 1  ( ) 2 2 4 4 xy x y x y dx dy    1 1  2 2 2 b dx dy a dx dy y x           ( ) 2 ( ) 0 2 2 2 2 2            y b dx dy xy dx dy x a 1 2  3 / 2 2 2 2 1                 dx dy dx d y  1 0 3 2 2 2 2                          dx dy dx d y 2 2 5.1 EQUATIONS WITH SEPARABLE VARIABLE Differential equations of the form f(x, y) dx dy  can be reduced to form ORDER AND DEGREE OF DIFFERENTIAL EQUATION 4 SOLUTION OF A DIFFERENTIAL EQUATION 5
g (x) h(y ) dx dy  where it is possible to take all terms involving x and dx on one side and all terms involving y and dy to the other side, thus separating the variables and integrating. Illustration 2 Question: Solve the differential equation x y y e x e dx dy     2 Solution: Separating the variables ( ) 2 e e x dx dy y x    e dy e x dx y x ( ) 2   , integrating, the solution is A x e e y x    3 3  e e x C y x    3 3 ( ) (C is an arbitrary constant) Illustration 3 Question: Find the order and degree of differential equation of all the parabolas whose axes are parallel to the x-axis and having a latus rectum a. Solution: Equation of required parabola’s is (y  ) 2 = a (x  ) Differentiating both sides w.r.t. x  a dx dy 2 (y  )  Again differentiating, w.r.t. x 2( ) 2 0 2 2 2           dx dy dx d y y  2( ) 2 0 3 2 2                 dx dy dx d y dx dy y  2 0 3 2 2         dx dy dx d y a Thus order of differential equation is 2 and degree is 1. 5.2 EQUATIONS REDUCIBLE TO EQUATIONS WITH SEPARABLE VARIABLE A differential equations of the form f (ax by c) dx dy    can not be solved by separating the variables directly. By substituting ax + by + c = t and dx dt dx dy a  b  , the differential equation can be separated in terms of variables x and t. Illustration 4 Question: Solve the differential equation cos(x y) dx dy   Solution: Put x + y = t  dx dt dx dy 1  Thus t dx dt 1  cos  dx t dt  1 cos Integrating both sides