Differential equations and numerical solutions · Ecuaciones diferenciales y soluciones numéricas
| English | Español |
|---|---|
| initial condition/ɪˈnɪʃl kənˈdɪʃn/ | condición 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? · ¿Cuál descripción define correctamente la condición inicial?
A specified value that selects a particular solution from a family. · Un valor especificado que selecciona una solución particular de una familia.
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 · Ecuaciones diferenciales y soluciones numéricas
For dy/dx=ky, separate variables: dy/y=k dx, giving y=Ae^(kx) · Para dy/dx=ky, separen variables: dy/y=k dx, obteniendo y=Ae^(kx)
Compare the model with the worked case and explain one change. · Compara el modelo con el caso resuelto y explica un cambio.
Using Euler with h=0.2 for y prime=0.5y,y₀=2, find y₁. · Usando Euler con h=0.2 para y'=0.5y, y₀=2, encuentre 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 esa aproximación, encuentre y₂.
Repeat from y₁: y₂=2.2+0.2×0.5×2.2=2.42. · Repita desde y₁: y₂=2.2+0.2×0.5×2.2=2.42.
Every step size gives exactly the same Euler approximation. · Cada tamaño de paso proporciona exactamente la misma aproximación 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. · Una condición inicial determina la constante de integración; no reemplaza a la integración. Dividir por y puede omitir una solución de equilibrio y=0. La precisión de Euler depende del tamaño de paso y de la ecuación.
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), encuentre y en x=0.
At x=0 the exponential is 1, so y=2. · En x=0 el exponencial es 1, por lo tanto y=2.
Match each part of a complete solution to its purpose. · Emparejar cada parte de una solución completa con su propósito.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Una suposición justifica el modelo; una comprobación verifica el resultado; la interpretación lo conecta con la pregunta.
Use this in your course
- edexcel IAL pure 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.