Q. 7 - Calculus

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 a differential equation and what it means for a function to be a solution of a differential equation

Step-by-Step Solution

Answer

A differential equation is an equation with one or more derivatives of a function.

A solution is a function $y=f\left(x\right)$ that satisfies the differential equation when $f$ and its derivatives are substituted into the equation.

Step 1. Given Information.

The objective is to write the definition of a differential equation and what it means for a function to be a solution of a differential equation.

Step 2. Definition of Differential Equation.

A differential equation is an equation with one or more derivatives of a function. It can also be defined as the equation that contains derivatives of one or more dependent variables with respect to one or more independent variables. For example,

$\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}=6y$ is a differential equation of the function,

$y=e^{-3x}$.

Step 3. Solution of Differential Equation.

A differential equation consists of a function $y=f\left(x\right)$ and one or more of its derivatives. A solution is a function $y=f\left(x\right)$ that satisfies the differential equation when $f$ and its derivatives are substituted into the equation.