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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.

ምሳሌ · 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

እርምጃ በደረጃ

  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.

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

አሁን ችግርን ይምረጡ፣ ወይም ጻፉ ወይም በራሳችሁ ቅርጽ አቅርቡ። እርምጃው ሁሉ፣ ምስል፣ መልሱ እስከምትጠይቅ ድረስ የተደበቀ ነው።

የራስዎን ስራዎች ይያዙ

ነጻ የሆኑት መተግበሪያዎች በየክፍል ውስጥ ማስታወሻዎችን ያካትታሉ፣ የተጠናቀቁትን ነገሮች መዝገብ፣ የተፈቱ ችግሮችን በአንድ ቦታ፣ ስለዚህ ገጽ መጠየቅ የሚችሉትን ረዳት ያካትታሉ፡፡ የቁጥር ትምህርት ለሁሉም ሰው ክፍት ነው፣ ተቀባይነት ያለው ወይም የለም፤

ምዝገባ መግባቱ

ፊደላት

ለሙሉ መግለጫ፣ ምስል፣ እና በውስጡ ያለውን ፊደል ትርጓሜ ለማየት ማንኛውንም ምልክት ይጫኑ።

እንዴት: 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.

ጥያቄዎች

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.

በ Differential Equations

Separable equationsFirst-order linear equationsNonhomogeneous equationsModelling with differential equations