Differential equations and numerical solutions
| English | 中文 | Pinyin |
|---|---|---|
| initial condition/ɪˈnɪʃl kənˈdɪʃn/ | 初始条件 | chū shǐ tiáo jiàn |
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
- For dy/dx=ky, separate variables: dy/y=k dx, giving y=Ae^(kx). Use an initial condition to find A. Euler's method takes y next=y+h f(x,y), with a chosen step h.
- 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?
A specified value that selects a particular solution from a family.
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.
With dy/dx=0.5y and y(0)=2, y=2e^(0.5x). Euler with h=0.2 gives y(0.2)≈2+0.2×1=2.2 and y(0.4)≈2.2+0.2×1.1=2.42. The exact second value is about 2.4428.
Differential equations and numerical solutions
For dy/dx=ky, separate variables: dy/y=k dx, giving y=Ae^(kx)
Compare the model with the worked case and explain one change.
Using Euler with h=0.2 for y prime=0.5y,y₀=2, find y₁.
Euler uses y₁=y₀+h f(y₀)=2+0.2×0.5×2=2.2.
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.
Every step size gives exactly the same Euler approximation. This claim is false. Explain which definition or assumption it violates.
For that approximation, find y₂.
Repeat from y₁: y₂=2.2+0.2×0.5×2.2=2.42.
Every step size gives exactly the same Euler approximation.
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.
Interpret a new situation
- For second-order linear equations, combine the complementary function with an appropriate particular integral, then apply the required initial conditions. This is Further Pure scope; check the named unit.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For y=2e^(0.5x), find y at x=0.
At x=0 the exponential is 1, so y=2.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.