Creating Differential Equations Solution1

In this page creating differential equations solution1 we are going to see solutions of some practice questions.

Form a differential equations by eliminating arbitrary constants given in brackets against each.

(i) y² = 4ax       {a}

Solution:

here we have only one arbitrary constant,so we can differentiate the given equation with respect to x.

    2y (dy/dx) =4a (1)

    2y (dy/dx) =4a

        dy/dx = 4a/2y

        dy/dx = 2a/y  ---- (1)

now we are going to find the value of "a" in terms of x and y from the given question.

         y² = 4ax

         a = y²/4x

now we are going to apply the value of a in the first equation

        dy/dx = 2(y²/4x)/y

        dy/dx = 2y²/4xy

             y' = y/2x

          2 x y' = y

           y = 2 x y'

Therefore the required equation is y = 2 x y'.


(ii) y = a x² + bx + c       {a,b}

Solution:

here we have two arbitrary constants,so we can differentiate the given equation with respect to x two times

    y = a x² + bx + c   --- (1)

differentiate with respect to x

   dy/dx = a (2 x) + b (1) + 0

   dy/dx = 2 a x + b

again differentiate the above equation with respect to x

    d²y/dx² = 2 a (1) + 0

     d²y/dx² = 2 a 

     y' = 2 a x + b     --- (2)

     y'' = 2 a    --- (3)

 we are going to apply the value of 2a in the second equation

 y' = y'' x + b

  b = y' - y'' x

  a = y''/2

 now we are going to apply the values of a and b in the first equation 

    y = a x² + b x + c

    y = (y''/2)x² + (y' - y'' x)x + c

    y = (y''x²/2) + y' x - y'' x² + c

    y = (y''x² + 2 y' x - 2 y'' x² + 2 c)/2

  2 y = 2 y' x - y'' x² + 2 c

  x² y'' - 2 x y' + 2 y - 2 c = 0

Therefore the required differential equation is x² y'' - 2 x y' + 2 y - 2 c = 0

creating differential equations solution1 creating differential equations solution1