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Nonhomogeneous equations

Driving forces: homogeneous solution plus a particular solution.

With a forcing term on the right, the general solution is the homogeneous solution plus any one particular solution. Undetermined coefficients guesses the particular solution in the shape of the forcing (exponential, polynomial, sinusoid); variation of parameters handles the rest.

ምሳሌ · y'' - 3y' + 2y = e^x

Differential-Equations · Differential-Equations

Y'' - 3y' + 2y = e^x

2 y{\left(x \right)} - 3 \frac{d}{d x} y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = e^{x}

እርምጃ በደረጃ

  1. 2 y{\left(x \right)} - 3 \frac{d}{d x} y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = e^{x}

    The differential equation.

  2. \text{order } 2

    Order 2: the highest derivative present.

  3. \text{Undetermined coefficients}

    Solve the homogeneous part from the characteristic equation, then guess a particular solution shaped like the right-hand side.

  4. r^{2} - 3 r + 2 = 0

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

  5. r = 2,\ r = 1

    Its roots.

  6. y{\left(x \right)} = \left(C_{1} + C_{2} e^{x} - x\right) e^{x}

    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)} = \left(C_{1} + C_{2} e^{x} - x\right) e^{x}

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: Nonhomogeneous equations

  1. The differential equation.
  2. Order 2: the highest derivative present.
  3. Solve the homogeneous part from the characteristic equation, then guess a particular solution shaped like the right-hand side.
  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 equationsSecond-order, constant coefficientsModelling with differential equations