S3

Page 1: Introduction to ODEs

  • Modelling with 1st order ODEs

    • Concentration of substances, population of organisms, or temperature of coffee changes with time (t).

    • Initial value at time t = t0: p(t0) = P0.

    • Goal: Predict future values p(t) for t > t0.

Page 2: Finding p(t)

  • Use the growth or decay rate of P.

  • Let k(p,t) be the change in P over a small time interval (St).

    • Formula: p(t + St) = p(t) + k(p,t)St.

  • Rearranging gives: p(t + St) - p(t) = k(p,t)St.

  • In the limit as St approaches 0: dp = k(p,t) dt.

    • Initial condition: P(t0) = P0.

    • Resulting in an Initial Value Problem (IVP): P' = k(P,t).

Page 3: Understanding k(P,t)

  • k(P,t) is the rate of growth or decay of P(t).

    • If k > 0, P will grow; if k < 0, P will decay.

  • Assumptions are made to determine k(P,t).

  • After solving IVP, compare model solutions to real observations.

    • Evaluate if the solution is sensible; modify assumptions if necessary.

Page 4: Solving Simple ODEs and IVPs

  • Example 1:(a) General solution for dy/dt = t.(b) Solve the initial value problem: dy = t dt, y(0) = 1.

    • Solution (a): dt = J(t) + C.

    • Rearrangement leads to y(t) = C.

    • For IVP: y(0) = 1 leads to C = 1.

    • Thus, y(t) = 1 + t^2/2.

Page 5: Graphs of Solutions

  • Illustrative graphs showing solutions with points (0, 0) and (to, yo).

  • Example of a function J(t) = 7.2.

Page 6: More ODE Solutions

  • Example 2:

    • General solution to ODE J'(x) = 2e^x + x sin(t).

  • Suggested homework to integrate and find the solution on Moodle.

Page 7: Advanced IVP

  • Example 3: Solve IVP: dy/dt = t - 1 (t)e^{t-1}, with y(1) = 2e.

  • Steps to derive solution involve differentiation and substitution.

Page 8: Initial Value Problem Solutions

  • Example 4: IVP: dy/dt = e^t, y(0) = 1.

  • Solution process involves integrating and applying the chain rule.

  • Final IVP solution derived: y(t) = e^{t}.

Page 9: Explicit vs Implicit Solutions

  • Remarks: Some equations yield implicit solutions.

  • Example: x - y - y^3 = 0 is an implicit solution of an ODE.

Page 10: The Inverse Problem

  • Question: Given a function y=f(x, C1, ..., Cu), find the ODE with this as a general solution.

  • Answer: Differentiate the function n-times to eliminate parameters.

  • Example includes deriving and manipulating specific parametric equations.

Page 11: Steps to Solve Inverse Problems

  • Clarifying steps for solving y=C1e^x + C2e^{2x}.

  • Each step involves differentiating and eliminating constants.

Page 12: Homework Assignment

  • Homework: Find the ODE that admits the implicit general solution x - y - y^3 = C1.