(a) By inspection find a one-parameter family of solutions...

(a) By inspection find a one-parameter family of solutions of the differential equation $x y^{\prime}=y .$ Verify that each member of the family is a solution of the initial-value problem \[ x y^{\prime}=y, y(0)=0 \] (b) Explain part (a) by determining a region $R$ in the $x y$ -plane for which the differential equation $x y^{\prime}=y$ would have a unique solution through a point $\left(x_{0}, y_{0}\right)$ in $R$ (c) Verify that the piecewise-defined function \[ y=\left\{\begin{array}{ll} 0, & x<0 \\ x, & x \geq 0 \end{array}\right. \] satisfies the condition $y(0)=0 .$ Determine whether this function is also a solution of the initial-value problem in part (a).

(a) By inspection find a one-parameter family of solutions of the differential equation $x y^{\prime}=y .$ Verify that each member of the family is a solution of the initial-value problem
\[
x y^{\prime}=y, y(0)=0
\]
(b) Explain part (a) by determining a region $R$ in the $x y$ -plane for which the differential equation $x y^{\prime}=y$ would have a unique solution through a point $\left(x_{0}, y_{0}\right)$ in $R$
(c) Verify that the piecewise-defined function
\[
y=\left\{\begin{array}{ll}
0, & x<0 \\
x, & x \geq 0
\end{array}\right.
\]
satisfies the condition $y(0)=0 .$ Determine whether this function is also a solution of the initial-value problem in part (a).

Key Concepts

One-Parameter Family of Solutions

This concept refers to the set of solutions to an ordinary differential equation (ODE) that includes an arbitrary constant, representing the degree of freedom due to integration. In the case of first order ODEs, the general solution typically contains one arbitrary constant, and each specific choice of the constant yields a distinct solution curve that satisfies the differential equation.

Initial Value Problem (IVP)

An initial value problem consists of a differential equation together with an initial condition that specifies the value of the unknown function at a particular point. This condition is used to select one particular solution from the general family of solutions. It is fundamental in applications because it provides a unique starting point for the evolution described by the ODE.

Existence and Uniqueness Theorem

This theorem, often associated with Picard–Lindelöf, provides conditions under which an initial value problem has a unique solution. It asserts that if the function on the right-hand side of the ODE and its partial derivative with respect to the dependent variable are continuous in a region around the initial point, then there exists a unique solution passing through that point. The theorem explains why certain regions in the plane guarantee unique behavior.

Singular Points in Differential Equations

Singular points are values where the differential equation fails to be well-behaved, often due to coefficients vanishing or becoming unbounded. In the context of an ODE, such as one where a coefficient multiplies the derivative, a singular point might lead to a breakdown of classical existence or uniqueness theorems. Special care must be taken when dealing with initial conditions at these points.

Piecewise-Defined Functions in Differential Equations

Piecewise-defined functions are those that have different expressions in different regions of their domain. In the context of differential equations, such functions are used to represent solutions that change behavior across a point or interval. The analysis involves checking not only that the function satisfies the differential equation in each region but also ensuring that the function meets the necessary continuity or differentiability conditions across the boundaries between regions.

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