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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.

වැඩ කළ උදාහරණය · 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}

පියවරෙන් පියවර

  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.

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

දැන් ඔයා ප් රශ්නයක් තෝරන්න, හෝ වර්ගය හෝ ඔබේම අඳින්න. සෑම පියවරක්, පින්තූරයක්, ඔබ ඉල්ලා තෙක් පිළිතුර සැඟවුණු.

ඔයාගෙ වැඩේ කරගෙන යන්න

නිදහස් ගිණුමක් සෑම පාඩමක් මත සටහන් එකතු, ඔබ අවසන් කර ඇති දේ වාර්තාවක්, එක් ස්ථානයක ඔබේ විසඳා ගැටළු, හා ඔබ මෙම පිටුව ගැන විමසීමට හැකි ගුරුවරයෙකු. ගණිතය ම සියලු දෙනාට විවෘත වේ, ඇතුලත් හෝ නැත.

ලියාපදිංචි වන්න පිවිසුම්

මෙහිදී භාවිත කරන සංකේත

සම්පූර්ණ අර්ථ දැක්වීම සඳහා ඕනෑම සංකේතයක් ටැප්, පින්තූරයක්, සහ එය සෑම අකුරු අදහස් කරන්නේ කුමක්ද.

කොහොමද: 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.

ජනතාව අහනවා ප්රශ්න

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.

තවත් Differential Equations

Separable equationsFirst-order linear equationsSecond-order, constant coefficientsModelling with differential equations