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Differential-Equations · Differential-Equations

Y'' - y = e^(2x)

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

Krok za krokom

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

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

  5. r = -1,\ r = 1

    Its roots.

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

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

  7. \checkmark

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

Odhaliť odpoveď
y{\left(x \right)} = C_{1} e^{- x} + C_{2} e^{x} + \frac{e^{2 x}}{3}