# CPT 02: ODEs

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

• Section 1.2 – Solutions and Initial Value Problems

## Expected Educational Results

• 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 Equations

### Definition: Differential Equation (DE)

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

### Definition: Ordinary Differential Equation (ODE)

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 $\frac{dy}{dx}=f\left(x,y\right)$$\dfrac{dy}{dx}=f(x,y)$.

### Definition: Partial Differential Equation (PDE)

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: $\frac{\partial u\left(t,x,y\right)}{\partial t}=D\left(\frac{{\partial }^{2}u\left(t,x,y\right)}{\partial {x}^{2}}+\frac{{\partial }^{2}u\left(t,x,y\right)}{\partial {y}^{2}}\right)$$\dfrac{\partial u(t,x,y)}{\partial t}=D\left(\dfrac{\partial^2u(t,x,y)}{\partial x^2}+\dfrac{\partial^2u(t,x,y)}{\partial y^2}\right)$.

NOTE: This course will only consider ODEs.

### Definition: Order of a DE

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

### Definition: Linear DE

Linear DE: A DE is linear if

1. the coefficients of the derivatives (including zeroth derivative) are functions of only the independent variable, and

2. 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.

### Question 01

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

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

2. The relation between air pressure, $p$$p$, and altitude, $y$$y$: $\frac{dp}{dy}=-\rho \left(y\right)g$$\displaystyle \frac{dp}{dy}=-\rho(y)g$

3. RLC circuit: $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$

4. 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$$\displaystyle \frac{1}{2}\sigma^2S^2\frac{\partial^2V}{\partial S^2}+rS \frac{\partial V}{\partial S}+\frac{\partial V}{\partial t}-rV=0$

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

6. Axial deformation on a bar: $\frac{d}{dx}\left[A\left(x\right)\sigma \left(x\right)\right]+f\left(x\right)=0$$\displaystyle \frac{d}{dx}\left[A(x)\sigma(x)\right]+f(x)=0$

7. 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$$\displaystyle \frac{d^2}{dx^2}\left(EI\frac{d^2\nu}{dx^2}\right)+k\nu=q$

8. Torsion of elastic bars: $\frac{{\partial }^{2}\theta }{\partial {x}^{2}}+\frac{{\partial }^{2}\theta }{\partial {y}^{2}}+2=0$$\displaystyle \frac{\partial^2\theta}{\partial x^2}+\frac{\partial^2\theta}{\partial y^2}+2=0$

9. Irrotational fluid flow: $\frac{{\partial }^{2}\varphi }{\partial {x}^{2}}+\frac{{\partial }^{2}\varphi }{\partial {y}^{2}}=0$$\displaystyle \frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}=0$

10. DEs to explore cancer-immune dynamics and tumor dormancy: $\frac{dC}{dt}=\frac{\mu }{\alpha }\left(1+\mathrm{\Psi }\left(I,C\right)\right)C\left(1-{\left(\frac{C}{{K}_{c}}\right)}^{\alpha }\right)$$\displaystyle \frac{dC}{dt}=\frac{\mu}{\alpha}\left(1+\Psi(I,C)\right)C\left(1-\left(\frac{C}{K_c}\right)^{\alpha}\right)$

11. Free fall with air resistance: $m\frac{dv}{dt}=mg-\frac{1}{2}\rho {C}_{D}A{v}^{2}$$\displaystyle m\frac{dv}{dt}=mg-\frac{1}{2}\rho C_DAv^2$

12. Decomposition of ${\text{NO}}_{2}$$\text{NO}_2$: $\frac{d\left[{\text{NO}}_{2}\right]}{dt}=-k\left[{\text{NO}}_{2}{\right]}^{2}$$\displaystyle \frac{d[\text{NO}_2]}{dt}=-k[\text{NO}_2]^2$

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

14. Black hole mass: $\frac{dM}{dt}=-\frac{k}{{M}^{2}}$$\displaystyle \frac{dM}{dt}=-\frac{k}{M^2}$

15. Solow-Swan model: $\frac{\partial k}{\partial t}=s{k}^{\alpha }-\delta k$$\displaystyle \frac{\partial k}{\partial t}=sk^{\alpha}-\delta k$

16. Bertalanffy equation: $\frac{dW}{dt}=\eta S-kV$$\displaystyle \frac{dW}{dt}=\eta S-kV$

17. Wave equation in one-dimension: $\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}$