Chapter 1, Problem 1RE

(a)

To determine

Whether the differential equation $\frac{dy}{dt}=y$ is an ordinary differential equation or a partial differential equation.

Answer to Problem 1RE

The differential equation $\frac{dy}{dt}=y$ is an ordinary differential equation.

Explanation of Solution

The ordinary differential is the equation in which the dependent variable is differentiated to only one independent variable.

The partial differential equation is the equation in which dependent variable is differentiated to more than one independent variable.

Here, the given dependent variable is differentiated to only one independent variable.

Thus, the differential equation $\frac{dy}{dt}=y$ is an ordinary differential equation.

(b)

To determine

The order of the differential equation $\frac{dy}{dt}=y$.

Answer to Problem 1RE

The order of the differential equation $\frac{dy}{dt}=y$ is 1.

Explanation of Solution

Formula used:

The standard form of the differential equation is given by $a_n\left(x\right)\frac{d^{n}y}{d{x}^n}+a_{n-1}\left(x\right)\frac{d^{n-1}y}{d{x}^{n-1}}+\cdots +a_1\left(x\right)\frac{dy}{dx}+a_o\left(x\right)y=f\left(x\right)$ where, $n$ is the order of the differential equation.

Calculation:

The order of the highest order differential, in a differential equation, is the order of the differential equation. Order is the number of times the dependent variable is differentiated with respect to independent variable.

Comparing with the standard equation the order of the differential is one.

Thus, the order of the differential equation $\frac{dy}{dt}=y$ is 1.

(c)

To determine

Whether the differential equation $\frac{dy}{dt}=y$ is linear or non-linear.

Answer to Problem 1RE

The differential equation $\frac{dy}{dt}=y$ is a linear differential equation.

Explanation of Solution

Formula used:

The standard form of linear differential is given by $a_o\left(x\right)y+a_1\left(x\right)\frac{dy}{dx}+a_2\left(x\right)\frac{d^{2}y}{d{x}^2}+\cdots +a_n\left(x\right)\frac{d^{n}y}{d{x}^n}+b\left(x\right)=0$.

Calculation:

The linear differential equation is the differential equation which is defined by a linear polynomial in the unknown function and its derivative.

By comparing with the standard equation, the given equation is linear differential equation. Degree of the differential equation is 1.

Thus, the differential equation $\frac{dy}{dt}=y$ is a linear differential equation.