Question

Form the differential equation having `y=(sin^(-1)x)^2+Acos^(-1)x+B ,w h e r eAa n dB` are arbitrary constants, as its general solution.

Answer 1

Given, ``y=(sin^(-1)x)^(2)+Acos^(-1)x+B
On differentiating w.r.t.x, we get
`(dy)/(dx)=(2sin ^(-1)x)/(sqrt1-x^(2))+(-A)/(sqrt(1-x^(2)))`
`Rightarrow sqrt(1-x^(2)) (dy)/(dx)=2 sin^(-1)x-A`
Again, differentiating w.r.t.x, we get
`sqrt(1-x^(2)) (d^(2)y)/(dx^(2))+(dy)/(dx). (-2x)/(2sqrt(1+x^(2)))=(2)/(sqrt(1-x^(2)))`
`Rightarrow (1-x^(2))(d^(2)y)/(dx^(2))-(x)/(sqrt(1-x^(2))).sqrt(1-x^(2))(dy)/(dx)=2`
`Rightarrow (1-x^(2))(d^(2)y)/(dx^(2))-x(dy)/(dx)=2`
`Rightarrow (1-x^(2))(d^(2)y)/(dx^(2))-x(dy)/(dx)-2=0`
Which is the required differential equation.