Differential Equations

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 9.2 – Direction Fields and Euler's Method

• Section 9.3 – Separable Equations

• Section 9.4 – Models for Population Growth

• Section 3.8 – Exponential Growth and Decay

Expected Educational Results

• Objective 15–01: I can identify ordinary differential equations (ODEs).

• Objective 15–02: I can identify the order of ODEs.

• Objective 15–03: I can solve first-order separable ordinary differential equations.

• Objective 15–04: I can solve first-order separable initial value problems.

• Objective 15–05: I can use technology to sketch a direction field for a first-order ordinary differential equation.

• Objective 15–06: I can model population growth and decay with a differential equation.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Differential Equations

Definition: Differential Equation [DE]

An equation that has any order derivative in any term.

Definition: Ordinary Differential Equation [ODE]

An equation that has any order ordinary derivative, e.g., ${\displaystyle \frac{d^{2}y}{d{x}^2}}$$\displaystyle \frac{d^2y}{dx^2}$, ${\displaystyle y^{\prime }(x)}$$\displaystyle y^{\,\prime}(x)$, etc., as a term or factor, where the ordinary derivative is the derivative of the dependent variable with respect to the only, i.e., one (1), independent variable.

Definition: Partial Differential Equation [PDE]

An equation that has any order partial derivative, e.g., ${\displaystyle \frac{{\partial }^{2}y}{\partial x\partial y}}$$\displaystyle \frac{\partial^2y}{\partial x\partial y}$, ${\displaystyle \frac{\partial y}{\partial t}}$$\displaystyle \frac{\partial y}{\partial t}$, etc., in any term, where the partial derivative is the derivative of the dependent variable with respect to one of the several independent variables.

NOTE: Partial derivatives are first discussed in Math 2215 Multivariate Calculus or Math 2551 Multivariate Calculus for Engineers. Thus, we will not discuss PDEs in this course.

Definition: Order of a DE

The order of a differential equation is the order of the highest-order derivative.

Investigation 01

Determine if the following DEs are ordinary. Explain.

1. ${\displaystyle y^{\prime }+e^{x}y=x^2}$$\displaystyle y^{\,\prime}+e^xy=x^2$

2. ${\displaystyle x{y}^{\prime \prime }+\ln (x)-x^2{y}^{\prime }=0}$$\displaystyle xy^{\,\prime\prime}+\ln{(x)}-x^2y^{\,\prime}=0$

3. ${\displaystyle y+\sin (x)=x^3{y}^{\prime \prime \prime }}$$\displaystyle y+\sin{(x)}=x^3y^{\,\prime\prime\prime}$

4. ${\displaystyle y^{\prime }+\cos (y)=\tan (x)}$$\displaystyle y^{\,\prime}+\cos{(y)}=\tan{(x)}$

5. Exponential growth and decay: ${\displaystyle \frac{dx}{dt}=kx}$$\displaystyle \frac{dx}{dt}=kx$

6. Simple harmonic motion: ${\displaystyle \frac{d^{2}x}{d{t}^2}=-\frac{k}{m}x}$$\displaystyle \frac{d^2x}{dt^2}=-\frac{k}{m}x$

7. RLC circuit: ${\displaystyle L\frac{d^{2}I}{d{t}^2}+R\frac{dI}{dt}+\frac{1}{C}I=0}$$\displaystyle L\frac{d^2I}{dt^2}+R\frac{dI}{dt}+\frac{1}{C}I=0$

8. Wave equation in one-dimension: ${\displaystyle \frac{{\partial }^{2}u}{\partial t^2}=c^2\frac{{\partial }^{2}u}{\partial x^2}}$$\displaystyle \frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}$

9. Torricelli's Law: ${\displaystyle \frac{dV}{dt}=-k\sqrt{V}}$$\displaystyle \frac{dV}{dt}=-k\sqrt{V}$

10. Newton's Second Law: ${\displaystyle {\overrightarrow{F}}_{net}=m\frac{d^2\overrightarrow{x}}{d{t}^2}}$$\displaystyle \vec{F}_{net}=m\frac{d^2\vec{x}}{dt^2}$

Check Your Answers

Investigation 02

Determine the order of the DE. Explain.

1. ${\displaystyle y^{\prime }+e^{x}y=x^2}$$\displaystyle y^{\,\prime}+e^xy=x^2$

2. ${\displaystyle x{y}^{\prime \prime }+\ln (x)-x^2{y}^{\prime }=0}$$\displaystyle xy^{\,\prime\prime}+\ln{(x)}-x^2y^{\,\prime}=0$

3. ${\displaystyle y+\sin (x)=x^3{y}^{\prime \prime \prime }}$$\displaystyle y+\sin{(x)}=x^3y^{\,\prime\prime\prime}$

4. ${\displaystyle y^{\prime }+\cos (y)=\tan (x)}$$\displaystyle y^{\,\prime}+\cos{(y)}=\tan{(x)}$

5. Exponential growth and decay: ${\displaystyle \frac{dx}{dt}=kx}$$\displaystyle \frac{dx}{dt}=kx$

6. Simple harmonic motion: ${\displaystyle \frac{d^{2}x}{d{t}^2}=-\frac{k}{m}x}$$\displaystyle \frac{d^2x}{dt^2}=-\frac{k}{m}x$

7. RLC circuit: ${\displaystyle L\frac{d^{2}I}{d{t}^2}+R\frac{dI}{dt}+\frac{1}{C}I=0}$$\displaystyle L\frac{d^2I}{dt^2}+R\frac{dI}{dt}+\frac{1}{C}I=0$

8. Torricelli's Law: ${\displaystyle \frac{dV}{dt}=-k\sqrt{V}}$$\displaystyle \frac{dV}{dt}=-k\sqrt{V}$

9. Newton's Second Law: ${\displaystyle {\overrightarrow{F}}_{net}=m\frac{d^2\overrightarrow{x}}{d{t}^2}}$$\displaystyle \vec{F}_{net}=m\frac{d^2\vec{x}}{dt^2}$

Check Your Answers

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Created: Thursday, 15 October 2020 6:42 EDT Last Modified: Thursday, 14 July 2022 - 08:51 (EDT)