Differential equations and numerical solutions · Équations différentielles et solutions numériques
| English | Français |
|---|---|
| initial condition/ɪˈnɪʃl kənˈdɪʃn/ | condition initiale |
What does the starting value decide?
- A growing population's rate is proportional to its current size. The rate equation describes a whole family until an initial population is supplied.
- This lesson studies initial condition 初始条件: A specified value that selects a particular solution from a family.
Choose the mathematical structure
- Separate the variables in dy/dx=ky, integrate both sides and apply an initial condition. Include any equilibrium solution excluded when dividing by y.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines initial condition? · Quelle description définit correctement une condition initiale ?
A specified value that selects a particular solution from a family. · Une valeur spécifiée qui sélectionne une solution particulière à partir d'une famille.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For dy/dx=0.5y and y(0)=2, separation gives ln|y|=0.5x+C, hence y=Ae^(0.5x). The initial value gives A=2. Differentiate the result to check the equation and substitute x=0 to check the initial condition.
Differential equations and numerical solutions · Équations différentielles et solutions numériques
Separate the variables in dy/dx=ky, integrate both sides and apply an initial condition · Séparez les variables dans dy/dx=ky, intégrez les deux côtés et appliquez une condition initiale
Compare the model with the worked case and explain one change. · Compare le modèle avec l'exemple résolu et explique un changement.
For y=2e^(0.5x), find y(0). · Pour y=2e^(0.5x), trouvez y(0).
At x=0, e⁰=1, so y=2. · En x=0, e⁰=1, donc y=2.
Test a tempting shortcut
- An initial condition determines the integration constant; it is not a replacement for integrating. Dividing by y can omit an equilibrium solution y=0. Euler accuracy depends on step size and the equation.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
An initial condition never changes the constant in a differential-equation solution. This claim is false. Explain which definition or assumption it violates.
For that solution, find the initial derivative. · Pour cette solution, trouvez la dérivée initiale.
Differentiate y=2e^(0.5x): y prime=e^(0.5x). At zero it is 1. · Différenciez y=2e^(0.5x) : y prime=e^(0.5x). En zéro, c'est 1.
An initial condition never changes the constant in a differential-equation solution. · Une condition initiale ne change jamais la constante dans une solution d'équation différentielle.
An initial condition determines the integration constant; it is not a replacement for integrating. Dividing by y can omit an equilibrium solution y=0. Euler accuracy depends on step size and the equation. · Une condition initiale détermine la constante d'intégration ; elle ne remplace pas l'intégration. Diviser par y peut omettre une solution d'équilibre y=0. La précision d'Euler dépend du pas et de l'équation.
Interpret a new situation
- This unit uses first-order separation and initial conditions. Euler numerical integration and second-order complementary functions are excluded from this lesson.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For that solution, find y divided by e when x=2. · Pour cette solution, trouvez y divisé par e lorsque x=2.
At x=2, y=2e, hence y/e=2. · En x=2, y=2e, donc y/e=2.
Match each part of a complete solution to its purpose. · Associer chaque partie d'une solution complète à son but.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Une hypothèse justifie le modèle ; une vérification teste le résultat ; l'interprétation le relie à la question.
Use this in your course
- edexcel IAL pure mathematics; official unit P4. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A specified value that selects a particular solution from a family. Choose the relationship, show the method, check its assumptions and interpret the result.