Overview
This lecture covers separable first order differential equations, how to recognize and solve them, and applies these techniques to real-world modeling problems, including population growth, cooling, and mixture problems.
Separable Differential Equations
- A first order equation is separable if it can be written as dy/dx = f(x)g(y).
- To solve, rearrange terms: 1/g(y) dy = f(x) dx, then integrate both sides.
- After integrating, combine constants of integration on each side into a single constant.
- Solve for y to obtain explicit solutions if possible.
Example: Factoring and Solving
- Given dy/dx = xy + x + y + 1, factor by grouping to get (y+1)(x+1).
- Separate variables: 1/(y+1) dy = (x+1) dx.
- Integrate: ln|y+1| = ½ x² + x + C.
- Solve for y: y = K e^(½ x² + x) - 1 where K is a constant.
Example: Exponentials and Logs
- Given x dy/dx = e^(-y), rewrite as e^y dy = (1/x) dx.
- Integrate: e^y = ln|x| + C.
- Solve: y = ln(ln|x| + C).
Example: Partial Fractions
- Given dx/dt = t/(x²-1), use partial fractions: 1/(x²-1) = -½/(x+1) + ½/(x-1).
- Integrate: ln|(x-1)/(x+1)| = t² + C.
- Solve for x and use initial condition to find the constant.
Population Growth Modeling
- Growth at a rate proportional to amount: dp/dt = kP.
- Separate and integrate: ln|P| = kt + C → P = Ce^(kt).
- Use given values (e.g., P(1), P(2)) to solve for C and k.
- Model: P = C e^(kt) with specific constants.
Newton’s Law of Cooling
- Temperature change proportional to difference from ambient: dT/dt = k(a - T).
- Separate and integrate: ln|T - a| = -kt + C → T = a + Ce^(-kt).
- Use initial temperature data points to solve for a (ambient temp), C, and k.
Mixture Problems
- Amount rate: da/dt = rate_in - rate_out (often da/dt = -k a for draining).
- Write the rate equations with appropriate units (e.g., pounds per minute).
- Solve: a = Ce^(-kt), use initial amount to find C.
Key Terms & Definitions
- Separable Equation — A differential equation that can be rewritten so all y terms are on one side and x terms on the other.
- Partial Fractions — Technique to split rational expressions for easier integration.
- Newton’s Law of Cooling — The rate of temperature change is proportional to the difference from ambient temperature.
- Initial Value Problem — Differential equation with specified value at a given point.
Action Items / Next Steps
- Practice separating and integrating several example equations, including those with partial fractions.
- Complete exercise: Solve for ambient temperature in Newton’s Law of Cooling using the given system of equations.