미분 방정식 및 초기 조건
도입됨| English | 의미 |
|---|---|
| initial condition/ɪˈnɪʃl kənˈdɪʃn/ | 초기 조건 |
| characteristic root/ˌkærɪktəˈrɪstɪk ruːt/ | 특징근 |
A decision before an answer
- The same differential equation describes infinitely many motions until initial conditions select one.
- Your goal: Solve separable first-order equations.
Read the relationship
- For y′=ky, separation and integration give y=Ce^(kt); recover the zero solution if division by y was used. An initial value determines C.
- Solve constant-coefficient second-order equations.
For y′=2y and y(0)=3, the solution is:
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
The initial condition fixes C=3.
Use the defining rule
- A linear second-order constant-coefficient equation uses a characteristic polynomial. Distinct roots yield exponentials; a repeated root requires (C1+C2t)e^(rt).
- Use initial conditions and uniqueness conditions.
For a repeated root r=−1, the general solution includes:
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
The factor t supplies a second independent solution.
Check the conditions
- Complex roots α±iβ give e^(αt)(C1 cos βt+C2 sin βt). The real motion includes both amplitude and phase information.
- Use initial conditions and uniqueness conditions.
Solve y″+4y=0 with y(0)=3 and y′(0)=4. The characteristic roots are ±2i, so y=A cos2t+B sin2t. The first condition gives A=3; differentiating gives y′(0)=2B=4, so B=2.
For y′=3y and y(0)=2, C = ____.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
At t=0 the exponential is 1.
Apply the task format
- Existence and uniqueness depend on hypotheses near the initial point. A singular coefficient or a failure of local Lipschitz behaviour can defeat the familiar uniqueness conclusion.
- Use initial conditions and uniqueness conditions.
A repeated characteristic root needs the factor t; two copies of the same exponential are not independent solutions.
Which answer fits this case?
Solve separable first-order equations
Initial conditions can select a unique solution when the relevant uniqueness hypotheses hold.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
Check the hypotheses rather than assuming uniqueness at a singular point.
Keep the distinctions
- initial condition 初始条件 — A value of the solution or derivative specified at a starting point.
- characteristic root 特征根 — A root of the polynomial governing exponential solutions.
- Solve separable first-order equations.
- Solve constant-coefficient second-order equations.
- Use initial conditions and uniqueness conditions.
Match each term with its precise meaning in this lesson.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
난이도 3/5 · 200 코인 + 오늘 첫 완료 시 소액 보너스
The order follows the stated process; check each stage before the next.