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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.

ਕੰਮ ਉਦਾਹਰਨ: y'' - 3y' + 2y = e^x

Differential-Equations · Differential-Equations

Solve 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}

ਕਦਮ ਦਰ ਕਦਮ

  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.

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

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