Differential equations and numerical solutions · Equações diferenciais e soluções numéricas
| English | Português |
|---|---|
| initial condition/ɪˈnɪʃl kənˈdɪʃn/ | condição inicial |
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
- For dy/dx=ky, separate variables: dy/y=k dx, giving y=Ae^(kx). Use an initial condition to find A. Euler's method takes y next=y+h f(x,y), with a chosen step h.
- 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? · Qual descrição define corretamente condição inicial?
A specified value that selects a particular solution from a family. · Um valor especificado que seleciona uma solução particular de uma família.
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.
With dy/dx=0.5y and y(0)=2, y=2e^(0.5x). Euler with h=0.2 gives y(0.2)≈2+0.2×1=2.2 and y(0.4)≈2.2+0.2×1.1=2.42. The exact second value is about 2.4428.
Differential equations and numerical solutions · Equações diferenciais e soluções numéricas
For dy/dx=ky, separate variables: dy/y=k dx, giving y=Ae^(kx) · Para dy/dx=ky, separe as variáveis: dy/y=k dx, resultando em y=Ae^(kx)
Compare the model with the worked case and explain one change. · Compare o modelo com o caso resolvido e explique uma mudança.
Using Euler with h=0.2 for y prime=0.5y,y₀=2, find y₁. · Usando Euler com h=0.2 para y prime=0.5y,y₀=2, encontre y₁.
Euler uses y₁=y₀+h f(y₀)=2+0.2×0.5×2=2.2. · Euler usa y₁=y₀+h f(y₀)=2+0.2×0.5×2=2.2.
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.
Every step size gives exactly the same Euler approximation. This claim is false. Explain which definition or assumption it violates.
For that approximation, find y₂. · Para essa aproximação, encontre y₂.
Repeat from y₁: y₂=2.2+0.2×0.5×2.2=2.42. · Repita a partir de y₁: y₂=2.2+0.2×0.5×2.2=2.42.
Every step size gives exactly the same Euler approximation. · Cada tamanho de passo fornece exatamente a mesma aproximação de Euler.
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. · Uma condição inicial determina a constante de integração; não substitui a integração. Dividir por y pode omitir uma solução de equilíbrio y=0. A precisão de Euler depende do tamanho do passo e da equação.
Interpret a new situation
- For second-order linear equations, combine the complementary function with an appropriate particular integral, then apply the required initial conditions. This is Further Pure scope; check the named unit.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For y=2e^(0.5x), find y at x=0. · Para y=2e^(0.5x), encontre y em x=0.
At x=0 the exponential is 1, so y=2. · Em x=0 o exponencial é 1, logo y=2.
Match each part of a complete solution to its purpose. · Associe cada parte de uma solução completa ao seu propósito.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Uma suposição justifica o modelo; uma verificação testa o resultado; a interpretação conecta-o à pergunta.
Use this in your course
- edexcel IAL further mathematics; official unit FP2. 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.