Let y = y ( x ) be the solution of the differential equation, x 2 + 1 2   d y d x + 2 x ( x 2 + 1 ) y = 1 such that y 0 = 0 . If a   y 1 = π 32 , then the value of a is

Question

Let y = y ( x ) be the solution of the differential equation, x 2 + 1 2 &#160; d y d x + 2 x ( x 2 + 1 ) y = 1 such that y 0 = 0 . If a &#160; y 1 = &#960; 32 , then the value of a is

Aicharya · Accepted Answer

x 2 + 1 2 &#160; d y d x + 2 x x 2 + 1 y = 1 &#8658; d y d x + 2 x x 2 + 1 y = 1 ( x 2 + 1 ) 2 Which is a linear differential equation with integrating factor I.F. = e &#8747; 2 x x 2 + 1 d x = e l n ( x 2 + 1 ) = x 2 + 1 The solution of the given differential equation will be, y . x 2 + 1 = &#8747; x 2 + 1 . 1 x 2 + 1 2 d x = &#8747; d x x 2 + 1 = t a n - 1 x + c Now, y 0 = &#160; 0 &#8658; C = 0 and y = 1 ( x 2 + 1 ) t a n - 1 x a y ( 1 ) = &#960; 32 &#8658; a = &#960; / 32 &#960; / 8 = 1 4 &#8658; a = 1 16

Let y = y ( x ) be the solution of the differential equation, x 2 + 1 2   d y d x + 2 x ( x 2 + 1 ) y =…

Solution

Let y = y ( x ) be the solution of the differential equation, x 2 + 1 2 &#160; d y d x + 2 x ( x 2 + 1 ) y =…

Solution

Let y = y ( x ) be the solution of the differential equation, x 2 + 1 2 d y d x + 2 x ( x 2 + 1 ) y =…