differential equations

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

This is a separable equation:

 

y/√1-y2 dy= 2x dx

 

you integrate:

 

 

-ln(√1-y2)=x^2+k

 

and hence:

 

√1-y2=Cexp(-x^2)

 

you can see C=√1-y(0)^2

If |y(0)|>1, this problems does not have a solution which is the case for the given initial value.