Differential equations and numerical solutions
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| initial condition/ɪˈnɪʃl kənˈdɪʃn/ | điều kiện ban đầu |
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.
The method
- Separate the variables in dy/dx=ky, integrate both sides and apply an initial condition.
- Include any equilibrium solution excluded when dividing by y.
For dy/dx=0.5y with y(0)=2, what does the initial condition determine?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Separation gives the family y=Ae^(0.5x) with a free constant. Substituting x=0, y=2 fixes A=2, selecting one particular solution from the family.
A worked case
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).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
At x=0, e⁰=1, so y=2.
Where it goes wrong
- 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.
Not true: An initial condition never changes the constant in a differential-equation solution. Euler's method depends on the step size; smaller steps usually give a more accurate value.
For that solution, find the initial derivative.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
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.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Euler's method depends on the step size; smaller steps usually give a more accurate value.
Applying it
- This unit uses first-order separation and initial conditions.
- Euler numerical integration and second-order complementary functions are excluded from this lesson.
For that solution, find y divided by e when x=2.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
At x=2, y=2e, hence y/e=2.
Match each step to its purpose in solving dy/dx=ky.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Separation gathers each variable on one side; integration gives the family; the initial condition picks the particular solution.
In your course
- AQA A-level Mathematics (7357), specification section H.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Initial condition: A specified value that selects a particular solution from a family.
- Separate the variables in dy/dx=ky, integrate both sides and apply an initial condition.
- Watch for: an initial condition determines the integration constant; it is not a replacement for integrating.