CPT 02: ODEsExpected Educational ResultsDifferential EquationsDefinition: Differential Equation (DE)Definition: Ordinary Differential Equation (ODE)Definition: Partial Differential Equation (PDE)Definition: Order of a DEDefinition: Linear DEQuestion 01CC BY-NC-SA 4.0

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

**Section 1.2**– Solutions and Initial Value Problems

**Objective 2–1**: I can distinguish between ordinary differential equations (ODEs) and partial differential equations (PDEs).**Objective 2–2**: I can identify the order of ODEs.**Objective 2–3**: I can identify and explain when ODEs are linear.**Objective 2–4**: I can verify if a curve is a solution to an ODE.

**Differential Equation (DE)**: A model (i.e., equation) that contains derivatives (e.g., rate
of change, acceleration, etc.) of an unknown function.

**Ordinary Differential Equation (ODE)**: A DE that contains only ordinary derivatives
(i.e., the unknown function is a function of a *single independent variable*), usually written as

**Partial Differential Equation (PDE)**: A DE that contains only partial derivatives (i.e., the
unknown function is a function of *more than one independent variable*).

An example of a PDE is Fick's second law:

**NOTE**: This course will *only* consider ODEs.

**Order**: The **order** of a DE is the order of the highest-order derivative.

**Linear DE**: A DE is **linear** if

the coefficients of the derivatives (including zeroth derivative) are functions of

**only**the independent variable,**and**all of the derivatives (including zeroth derivative) are only to the first power.

If either of the conditions is **not** met, then the DE is **nonlinear**.

Identify the dependent variable, independent variable(s), and parameters in the following DEs:

Exponential growth and decay:

$\frac{dx}{dt}=kx$ The relation between air pressure,

, and altitude,$p$ :$y$ $\frac{dp}{dy}=-\rho (y)g$ RLC circuit:

$L\frac{{d}^{2}I}{d{t}^{2}}+R\frac{dI}{dt}+\frac{1}{C}I=0$ Black-Scholes equation:

$\frac{1}{2}{\sigma}^{2}{S}^{2}\frac{{\partial}^{2}V}{\partial {S}^{2}}+rS\frac{\partial V}{\partial S}+\frac{\partial V}{\partial t}-rV=0$ Newton's Second Law:

$\overrightarrow{F}}_{net}=m\frac{{d}^{2}\overrightarrow{x}}{d{t}^{2}$ Axial deformation on a bar:

$\frac{d}{dx}[A(x)\sigma (x)]+f(x)=0$ Differential equations to explore elastic beams:

$\frac{{d}^{2}}{d{x}^{2}}\left(EI\frac{{d}^{2}\nu}{d{x}^{2}}\right)+k\nu =q$ Torsion of elastic bars:

$\frac{{\partial}^{2}\theta}{\partial {x}^{2}}+\frac{{\partial}^{2}\theta}{\partial {y}^{2}}+2=0$ Irrotational fluid flow:

$\frac{{\partial}^{2}\varphi}{\partial {x}^{2}}+\frac{{\partial}^{2}\varphi}{\partial {y}^{2}}=0$ DEs to explore cancer-immune dynamics and tumor dormancy:

$\frac{dC}{dt}=\frac{\mu}{\alpha}(1+\mathrm{\Psi}(I,C))C(1-{\left(\frac{C}{{K}_{c}}\right)}^{\alpha})$ Free fall with air resistance:

$m\frac{dv}{dt}=mg-\frac{1}{2}\rho {C}_{D}A{v}^{2}$ Decomposition of

:${\text{NO}}_{2}$ $\frac{d[{\text{NO}}_{2}]}{dt}=-k[{\text{NO}}_{2}{]}^{2}$ Torricelli's Law:

$\frac{dV}{dt}=-k\sqrt{V}$ Black hole mass:

$\frac{dM}{dt}=-\frac{k}{{M}^{2}}$ Solow-Swan model:

$\frac{\partial k}{\partial t}=s{k}^{\alpha}-\delta k$ Bertalanffy equation:

$\frac{dW}{dt}=\eta S-kV$ Wave equation in one-dimension:

$\frac{{\partial}^{2}u}{\partial {t}^{2}}={c}^{2}\frac{{\partial}^{2}u}{\partial {x}^{2}}$

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**Last Modified**: Sunday, 23 August 2020 20:40 EDT