Question

On page 51 we showed that a one-parameter family of solutions of the first-order differential equation dy/dx=xy^(1/2) is y=((1/4)x^4+c)^2 for c>=0. Each solution in this family is defined on (-inf,inf). The last statement is not true if we choose c to be negative. For c=-1, explain why y=((1/4)x^4+c)^2 is not a solution of the DE on the interval (-inf,inf). Find an interval of definition I on which y=((1/4)x^4+c)^2 is a solution of the DE.

Answer 1

Note that

This is not a solution because the square root must be positive so we must have

for the equation to hold

But is also possible namely for this is true

So the interval of definition is and this must hold for all real x

So we must have

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