Chapter 1, Problem 1CR

To determine

To fill: The blank in the statement “$\frac{d}{dx}\left(c_1{e}^{10x}\right)=$ _______.

Answer to Problem 1CR

The completed statement is “$\frac{d}{dx}\left(c_1{e}^{10x}\right)=\underset{\_}{10y}$ where $y=c_1{e}^{10x}$.

Explanation of Solution

Procedure used:

Formation to a linear first order differential equation:

Begin with considering the case of a linear firs-order differential equation $\frac{d}{dx}\left(c{e}^{ax}\right)$, where $y=c_1{e}^{ax}$, $a$ is constant. Differentiate the linear firs-order differential equation $\frac{d}{dx}\left(c{e}^{ax}\right)$ becomes,

$\begin{array}{l}\frac{d}{dx}\left(c{e}^{ax}\right)=ac{e}^{ax}\\ \frac{dy}{dx}=ya\end{array}$

Calculation:

The given differential equation is, $\frac{d}{dx}{c}_1{e}^{10x}$.

Differentiate the differential equation $\frac{d}{dx}{c}_1{e}^{10x}$ with respect to x.

$\begin{array}{l}\frac{d}{dx}\left(c_1{e}^{10x}\right)=10c_1{e}^{10x}\\ \frac{dy}{dx}=10y\end{array}$

Hence, the completed statement is “$\frac{d}{dx}\left(c_1{e}^{10x}\right)=\underset{\_}{10y}$ where $y=c_1{e}^{10x}$.