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Differential-Equations · Differential-Equations

Y'' + 2y' + 5y = 0

5 y{\left(x \right)} + 2 \frac{d}{d x} y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0

Krok za krokom

  1. 5 y{\left(x \right)} + 2 \frac{d}{d x} 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} + 2 r + 5 = 0

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

  5. r = -1 - 2 i,\ r = -1 + 2 i

    Its roots.

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

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