Solve the differential equation y= 2xv1-y2

Question

a. Solve the differential equation y= 2x√1-y²

b. Explain why the initial value problem y =2x√1-y² with y(0) = 3 does not have a solution.

Answer 1

Assuming that the equation is

 

   dy/dx =   2 x sqrt(1 - y²)    the method of separation of variables will work. 

 

    dy /sqrt( 1- y²)  =  2 x dx     Integrate both sides

 

 

    sin^-1(y) =   x²  +   C   where   C is a constant of integration

 

  So  y = sin( x² +C)

 

    From this it can be seen the the absolute value of y must be less than or equal to 1.

   Therefore  y(0) = 3 is not possible.