Second-order linear differential equations · Ecuaciones diferenciales lineales de segundo orden
| English | Español |
|---|---|
| complementary function/ˌkɒmplɪˈmentəri ˈfʌŋkʃn/ | función complementaria |
Why do we need two initial conditions?
- A displacement model involves both velocity and acceleration. Solving only a first-order rate equation cannot capture both initial conditions.
- This lesson studies complementary function 互补函数: The general solution of the associated homogeneous linear differential equation.
Choose the mathematical structure
- For y double prime+ay prime+by=f(x), solve the auxiliary quadratic for the homogeneous part. Add a suitable particular integral for the forcing term. Repeated and complex roots require their correct forms.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines complementary function? · ¿Qué descripción define correctamente la función complementaria?
The general solution of the associated homogeneous linear differential equation. · La solución general de la ecuación diferencial lineal homogénea asociada.
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.
For y double prime-3y prime+2y=0, the auxiliary equation is m²-3m+2=0 with roots 1,2. Hence y=Ae^x+Be^(2x). With y(0)=1 and y prime(0)=0, A+B=1 and A+2B=0, so A=2,B=-1.
Second-order linear differential equations · Ecuaciones diferenciales lineales de segundo orden
For y double prime+ay prime+by=f(x), solve the auxiliary quadratic for the homogeneous part · Para y doble prima+ay prima+by=f(x), resuelva el cuadrático auxiliar para la parte homogénea
Compare the model with the worked case and explain one change. · Compara el modelo con el caso resuelto y explica un cambio.
For A+B=1 and A+2B=0, find A. · Para A+B=1 y A+2B=0, encuentre A.
Subtract A+B=1 from A+2B=0: B=-1, hence A=2. · Reste A+B=1 de A+2B=0: B=-1, luego A=2.
Test a tempting shortcut
- Two arbitrary constants need two independent conditions. If the trial particular integral duplicates a complementary-function term, multiply the trial by x as required. Do not discard a valid oscillatory solution because the auxiliary roots are complex.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
One initial value is always sufficient to determine both constants in a second-order general solution. This claim is false. Explain which definition or assumption it violates.
For those conditions, find B. · Para esas condiciones, encuentre B.
Subtract the first equation from the second to obtain B=-1. · Reste la primera ecuación de la segunda para obtener B=-1.
One initial value is always sufficient to determine both constants in a second-order general solution. · Un valor inicial siempre es suficiente para determinar ambas constantes en una solución general de segundo orden.
Two arbitrary constants need two independent conditions. If the trial particular integral duplicates a complementary-function term, multiply the trial by x as required. Do not discard a valid oscillatory solution because the auxiliary roots are complex. · Dos constantes arbitrarias requieren dos condiciones independientes. Si la integral particular de ensayo duplica un término de la función complementaria, multiplique la prueba por x según sea necesario. No descarte una solución oscilatoria válida porque las raíces auxiliares son complejas.
Interpret a new situation
- Differentiate the final expression and substitute into the original differential equation. Check both initial conditions separately, and interpret the permitted solution interval.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the larger root of m²-3m+2=0. · Encuentre la raíz mayor de m²-3m+2=0.
Factor m²-3m+2=(m-1)(m-2); the larger root is 2. · Factorice m²-3m+2=(m-1)(m-2); la raíz mayor es 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.
The general solution of the associated homogeneous linear differential equation. Choose the relationship, show the method, check its assumptions and interpret the result.