If y=y(x) is a particular solution of 1-x^2 d y d x + 2 x 1-x^2 y=x, y(0)=1 , then y ( 1 2 )=

Step-by-step solution

aligned & 1- x ^2 dy dx + 2 x 1- x ^2 y = x & dy dx + 2 x 1- x ^2 y = x 1- x ^2 .....(i) aligned Which is a linear differential equation I.f. = e ^ 2 x 1- x ^2 dx = e ^ - (1- x ^2 ) ; I.F = 1 1- x ^2 Now, the solution to the differential equation (i) is aligned & y. I.F. = x 1-x^2 1 1-x^2 d x+c & y 1 1-x^2 = 1 -2 -2 x (1-x^2 )^ 3 2 d x+c & y 1-x^2 =- 1 2 (1-x^2 )^ - 1 2 (- 1 2 ) +c & y= (1-x^2 ) [ (1-x^2 )^ -1 / 2 +c ].....(ii) & y(0)=1, then from eqn (ii), we get & y(0)=(1-0) [(1-0)^ -1 / 2 +c ] & 1=1+c c=0 aligne