Orthogonal trajectories

Given a family of curves ${C_k}$ where $k$ is a parameter that can be expressed by $x$ and $y$, an orthogonal trajectory of the family is a curve $C$ such that every intersection of $C$ and $C_i$ are perpendicular. To find $C$, we follow the following steps:

Step 1 Find $\frac{dy}{dx}=f(x,y)$ of $C_k$

Step 2 Replace $k$ in $f(x,y)$ in terms of $x$ and $y$.

Step 3 Let $\frac{dy}{dx} = -\frac{1}{f(x,y)}$

Step 4 Solve the last ODE.

Mixing Problem

A general mixing problem in calculus, often modeled by ordinary differential equations, involves describing the concentration or amount of a substance in a solution that is continuously stirred or mixed. These problems are common in various fields, such as chemistry, environmental science, and chemical engineering.

Here’s a typical setup for a mixing problem:

Problem Statement: Consider a tank $V$ containing a solution, and the volume of a substance is $A_0$. Let $C(t)$ represent the concentration of the substance in the tank at time $t$. The tank has an inlet through which a solution with concentration $A$ is flowing in at a rate of $R_a$ and an outlet through which the mixture is flowing out at a rate of $R_o$.

Key Variables:

• $V$: the volume of the tank.
• $A_0$: the initial volume of the substance.
• $C(t)$: Concentration of the substance in the tank at time $t$.
• $A$: Concentration of the incoming solution.
• $R_a$: Inflow rate.
• $R_o$: Outflow rate.

ODE Model: The general ODE representing this mixing problem is often expressed as:

with the initial condition $C(0)=\frac{A_0}{V}$.

where $V$ is the volume of the tank.

Strategy to Solve:

The above differential equation is of separable form. Indeed, we can rearrange the equation to