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
- 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?
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.
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
Separate the variables in dy/dx=ky, integrate both sides and apply an initial condition
Compare the model with the worked case and explain one change.
For y=2e^(0.5x), find y(0).
At x=0, e⁰=1, so 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.
Differentiate y=2e^(0.5x): y prime=e^(0.5x). At zero it is 1.
An initial condition never changes the constant in a differential-equation solution.
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
- 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.
At x=2, y=2e, hence y/e=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
- 7357 · A-level · H. Match the target tier and specification before assigning extensions.
- 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.