maths.free › Differential Equations › Second-order, constant coefficients

Second-order, constant coefficients

ay″ + by′ + cy = 0 and the characteristic equation: oscillation, damping, growth.

Try y = e^{rx} and the equation becomes a quadratic in r, the characteristic equation. Real distinct roots give exponentials, a repeated root adds an x, complex roots give sines and cosines. y″ + y = 0 is the simple harmonic oscillator: every pendulum, spring and LC circuit.

Exemple résolu · y'' + y = 0

Differential-Equations · Differential-Equations

Y'' + y = 0

y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0

Étape par étape

  1. y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0

    The differential equation.

  2. \text{order } 2

    Order 2: the highest derivative present.

  3. \text{Constant coefficients, homogeneous}

    Try y = e^{rx}: the characteristic polynomial in r gives the roots, and each root contributes a term to the general solution.

  4. r^{2} + 1 = 0

    Characteristic equation of the homogeneous part (substitute y = e^{rx}).

  5. r = - i,\ r = i

    Its roots.

  6. y{\left(x \right)} = C_{1} \sin{\left(x \right)} + C_{2} \cos{\left(x \right)}

    General solution (C₁, C₂ … are arbitrary constants).

  7. \checkmark

    Verified: substituting the solution back into the equation gives 0.

Révèle la réponse
y{\left(x \right)} = C_{1} \sin{\left(x \right)} + C_{2} \cos{\left(x \right)}

À vous Choisissez un problème, ou tapez ou dessinez le vôtre. Chaque étape, une image, la réponse cachée jusqu'à ce que vous le demandiez.

Conservez votre propre travail

Un compte gratuit ajoute des notes sur chaque leçon, un enregistrement de ce que vous avez terminé, vos problèmes résolus en un seul endroit, et un tuteur que vous pouvez demander sur cette page. Les maths eux-mêmes sont ouverts à tout le monde, signé ou non.

Inscription Connexion

Symboles utilisés ici

Appuyez sur n'importe quel symbole pour la définition complète, une image et ce que signifie chaque lettre.

Comment faire: Second-order, constant coefficients

  1. The differential equation.
  2. Order 2: the highest derivative present.
  3. Try y = e^{rx}: the characteristic polynomial in r gives the roots, and each root contributes a term to the general solution.
  4. Characteristic equation of the homogeneous part (substitute y = e^{rx}).
  5. Its roots.
  6. General solution (C₁, C₂ … are arbitrary constants).
  7. Verified: substituting the solution back into the equation gives 0.

Questions posées par les gens

What is a differential equation?

An equation whose unknown is a function, relating it to its own derivatives. "The rate of growth is proportional to the population" is y′ = ky, and solving it means finding y as a function of time.

Why does the solution have arbitrary constants?

Integrating loses information: many functions share the same derivative. An n-th order equation has n constants, fixed by n initial or boundary conditions.

Plus en Differential Equations

Separable equationsFirst-order linear equationsNonhomogeneous equationsModelling with differential equations