Differential equations and numerical solutions
도입됨| English | 의미 |
|---|---|
| initial condition/ɪˈnɪʃl kənˈdɪʃn/ | 초기 조건 |
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?
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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).
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
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.
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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.
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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.
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At x=2, y=2e, hence y/e=2.
Match each step to its purpose in solving dy/dx=ky.
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Separation gathers each variable on one side; integration gives the family; the initial condition picks the particular solution.
In your course
- Pearson Edexcel International A Level Mathematics, unit P4.
- 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.
Write from a starting value of 6
- y equals 6 e to the power 0.5 x.
- Read the value at x equals 0.
The solution y equals 6 e to the power 0.5 x comes from dy/dx equals 0.5 y. What is y at x equals 0?
The exponential factor is 1 at the start.
What is y at x equals 0?
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The initial condition is the value, not the slope. Other options — 2 (the lesson starting value); 3 (the initial slope).