Question

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible.

The definition of an initial-value problem and what it means for a function to be a solution of an initial-value problem.

Short answer

An initial value problem (IVP) is a differential equation along with an appropriate number of the initial conditions.

The solution to an initial value problem is a function that is a solution to the differential equation and satisfies the initial conditions.

Step-by-step solution

Step 1. Given Information

The objective is to define the initial-value problem and what it means for a function to be a solution to an initial-value function.

Step 2. Definition of Initial Value Problem.

An initial value problem (IVP) is a differential equation along with an appropriate number of the initial conditions. For example,

$2t\frac{dy}{dx}+4y=3,y\left(1\right)=-4$.

Step 3. Solution for an Initial Value Problem.

The solution to an initial value problem is a function that is a solution to the differential equation and satisfies the initial conditions.