MTH2032 Chap.2 Modelling with First-Order ODEs and IVPs
Modelling with First-Order ODEs and IVPs
Week 2 turns physical rules into differential equations and asks when an initial value problem has exactly one solution. Models are built by comparing a quantity at two close times and accounting for what enters or leaves, the balance technique that later produces the PDE models of Part II.
The existence-uniqueness theorems for scalar equations and for first-order systems give the conditions to check, and the conversion of higher-order equations into systems lets one theorem cover them all. The chapter also explains how to recognise when uniqueness can fail, using an example with two distinct solutions through the same point.
What this chapter covers
- 01
Balance between t and t + δt
- 02
Radioactive decay and half-life
- 03
Newton's law of cooling
- 04
Theorem 5.2 for scalar equations
- 05
Continuity checks without limits
- 06
Recasting to first-order systems
- 07
Theorem 6.2 for systems
Worked example · free
Half-life from a single measurement
- 1The solution of y′ = −ky is y = y(0)e^(−kt).
- 1Losing 10% in 3 days means e^(−3k) = 0.9.
- 1So k = −ln(0.9)/3 = 0.0351 per day.
- 1The half-life solves e^(−kt) = 0.5, so t = ln 2 / k = 19.7 days.
Key terms
- Balance Equation
- A statement that the change in a quantity over a short interval equals what is added minus what is removed.
- Half-Life
- The time for an exponentially decaying quantity to fall to half its value, equal to ln 2 divided by the decay rate.
- Rectangular Domain
- A region (a, b) × (c, d) of the xy-plane on which the hypotheses of the existence theorem are checked.
- Newton's Law of Cooling
- The rule that a body's temperature changes at a rate proportional to its difference from the surrounding temperature.
Modelling with First-Order ODEs and IVPs FAQ
How do I show that an initial value problem has a unique solution?
Write the equation as y′ = f(x, y), argue that f and its partial derivative in y are continuous on a rectangle, check that the starting point lies inside that rectangle, and then quote the fundamental existence-uniqueness theorem with its conclusion on an interval around the starting point. Each of those steps carries marks in the problem-set solutions.
Does the existence theorem say how long the solution lasts?
No. It promises a solution on a possibly short interval that contains the starting point. The notes give y′ = y² as an example where the solution exists only on part of the region where the right-hand side makes sense. Statements about the whole real line need a separate argument.
Why turn a third-order equation into a system?
Setting y1 = y, y2 = y′ and y3 = y″ produces a first-order system with the same information. The system form lets one theorem, Theorem 6.2, cover equations of any order, and it is also the starting point for numerical methods, which work with first-order systems.
How do I build a model from a word problem?
List each quantity with its units, decide what is gained and what is lost over a short time interval, and write that balance as an equation. Divide by the length of the interval and let it shrink to obtain the ODE. Then solve, apply the initial value, and finish with a sentence that interprets the answer in the physical setting.
Exam move
Practise modelling in three written layers: define each variable with its unit, write the balance over a short interval and take the limit, then solve and interpret. For decay and cooling models, learn to calibrate the rate constant from one measurement and to compute a half-life or halving time, and always state the interpretation in a sentence.
For existence questions, memorise the shape of the argument rather than particular functions: recast as y′ = f(x, y), cite the continuity rules for elementary functions, choose an explicit rectangle that avoids the problem points, place the data point inside it, and conclude with the interval.
Then practise the opposite skill: given an equation whose derivative in y blows up, explain why uniqueness can fail and exhibit two solutions. Finally, convert two or three higher-order equations into systems and state the initial vector each time, since the released verification problems ask for exactly this kind of short justification.
Close each modelling answer by checking the long-run behaviour of your formula against the physical situation, for example that a cooling body approaches room temperature.
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