Chapter 2.2, Problem 31E

Interval of Definition. By looking at an initial value problem $dy/dx=f\left(x,y\right)$ with $y\left(x_0\right)=y_0$, it is not always possible to determine the domain of the solution $y\left(x\right)$ or the interval over which the function $y\left(x\right)$ satisfies the differential equation.

a. Solve the equation $dy/dx=x{y}^3$.

b. Give explicitly the solutions to the initial value problem with $y\left(0\right)=1$; $y\left(0\right)=1/2$; $y\left(0\right)=2$.

c. Determine the domains of the solutions in part (b).

d. As found in part (c), the domains of the solutions depend on the initial conditions. For the initial value problem $dy/dx=x{y}^3$ with $y\left(0\right)=a$, $a>0$, show that as $a$ approaches zero from the right the domain approaches the whole real line $\left(-\infty ,+\infty \right)$ and as $a$ approaches $+\infty$ the domain shrinks to a single point.

e. Sketch the solutions to the initial value problem $dy/dx=x{y}^3$ with $y\left(0\right)=a$ for $a=\pm 1/2$, $\pm 1$, and $\pm 2$.