Differential equations and numerical solutions · 微分方程与数值解法
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| 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
- 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? · 以下哪项描述正确定义了初始条件？
A specified value that selects a particular solution from a family. · 从一族解中选定特定解的指定值。
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 · 微分方程与数值解法
Separate the variables in dy/dx=ky, integrate both sides and apply an initial condition · 分离 dy/dx=ky 的变量，对两边积分并应用初始条件。
Compare the model with the worked case and explain one change. · 对比模型与已解案例，并说明其中一处变化。
For y=2e^(0.5x), find y(0). · 已知 y=2e^(0.5x)，求 y(0)。
At x=0, e⁰=1, so y=2. · 当 x=0 时，e⁰=1，因此 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. · 对于该解，求初始导数。
Differentiate y=2e^(0.5x): y prime=e^(0.5x). At zero it is 1. · 对 y=2e^(0.5x) 求导：y' = e^(0.5x)。在零处的值为 1。
An initial condition never changes the constant in a differential-equation solution. · 初始条件不会改变微分方程解中的常数项。
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. · 初始条件用于确定积分常数；它不能替代积分过程。除以 y 可能会遗漏平衡解 y=0。欧拉法的精度取决于步长和方程本身。
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. · 对于该解，求当 x=2 时 y 除以 e 的值。
At x=2, y=2e, hence y/e=2. · 当 x=2 时，y=2e，因此 y/e=2。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
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.