On page 47 we showed that a one-parameter family of solutions of the first-order

Chapter 2, Problem 48

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On page 47 we showed that a one-parameter family of solutions of the first-order differential equation \(d y / d x=x y^{\frac{1}{2}}\) is \(y=\left(\frac{1}{4} x^{4}+c\right)^{2} \text { for } c \geq 0\). Each solution in this family is defined on the interval \((-\infty, \infty)\). The last statement is not true if we choose c to be negative. For c = -1, explain why \(y=\left(\frac{1}{4} x^{4}-1\right)^{2}\) is not a solution of the DE on \((-\infty, \infty)\). Find an interval of definition I on which \(y=\left(\frac{1}{4} x^{4}-1\right)^{2}\) is a solution of the DE.