Chapter 9, Problem 1CRQ

a.

To determine

To fill: The blank in the statement “An equation that involves an unknown function and its derivatives is called a/an ______” with the appropriate answer.

Answer to Problem 1CRQ

The completed statement is “An equation that involves an unknown function and its derivatives is called a $\underset{\_}{\text{differential equation}}$”.

Explanation of Solution

It is known that, “An equation which contains the derivative or differential of an unknown function is said to be a differential function”.

In other words, a differential equation is a relation between an unknown function and derivatives.

Consider a differential equation as, $\frac{{\partial }^{2}u}{\partial t^2}=c^2\frac{{\partial }^{2}u}{\partial x^2}$, where, $u\left(x,t\right)$ is the displacement from the rest position of the point x at time t and c is the speed of the wave.

That is, u is an unknown function that is a dependent variable and x, t are two independent variables.

Hence, “An equation that involves an unknown function and its derivatives is called a $\underset{\_}{\text{differential equation}}$”.

b.

To determine

To fill: The blank in the statement “A solution of a differential equation is any function that ____ the differential equation” with the appropriate answer.

Answer to Problem 1CRQ

The completed statement is “A solution of a differential equation is any function that $\underset{\_}{\text{satisfies}}$ the differential equation”.

Explanation of Solution

It is known that, “A solution of a differential equation $y^{\prime }$ is any function y that satisfies the differential equation”.

Consider a differential equation as, $y^{\prime }-y=0$, where $y=C{e}^x$.

In order to check whether the solution of the differential equation $y^{\prime }-y=0$ is $y=C{e}^x$, first differentiate $y=C{e}^x$ with respect to x.

$\begin{array}{c}{y}^{\prime }=\frac{d}{dx}\left(y\right)\\ =C\frac{d}{dx}\left(e^x\right)\\ =C{e}^x\end{array}$

Now substitute $y=C{e}^x$ and $y^{\prime }=C{e}^x$ in $y^{\prime }-y=0$ and check whether solution of the differential equation $y^{\prime }-y=0$ is $y=C{e}^x$.

$\begin{array}{l}C{e}^x-C{e}^x\stackrel{?}{=}0\\ 0=0\end{array}$

Observe that, the left hand side is equal to the right hand side.

Hence, “A solution of a differential equation is any function that $\underset{\_}{\text{satisfies}}$ the differential equation”.