Chapter 1.3, Problem 16P

To determine

(a)

To verify:

The hypothesis of the existence and uniqueness theorem are satisfied for the initial value problem $y^{\prime }=-2x{y}^2,y\left(x_0\right)=y_0$ for every $\left(x_0,y_0\right)$.

To determine

(b)

To verify:

All the values of the constant c, $y(x)=\frac{1}{\left(x^2+c\right)}$ is a solution to $y^{\prime }=-2x{y}^2$.

To determine

(c)

(i)

To find:

The maximum interval on which the solution is valid for the differential equation $y^{\prime }=-2x{y}^2$ and initial condition $y\left(0\right)=1$, and sketch the solution curve.

To determine

To find:

The maximum interval on which the solution is valid for the differential equation $y^{\prime }=-2x{y}^2$ and initial condition $y\left(1\right)=1$, and sketch the solution curve.

To determine

To find:

The maximum interval on which the solution is valid for the differential equation $y^{\prime }=-2x{y}^2$ and initial condition $y\left(0\right)=-1$, and sketch the solution curve.

To determine

(d)

To verify:

All the values of the constant c, $y(x)=\frac{1}{\left(x^2+c\right)}$ is a solution to $y^{\prime }=-2x{y}^2$.