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Differential-Equations · Differential-Equations

Y'' + 4y = 0

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

Step by step

  1. 4 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} + 4 = 0

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

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

    Its roots.

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

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

  7. \checkmark

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

Reveal the answer
y{\left(x \right)} = C_{1} \sin{\left(2 x \right)} + C_{2} \cos{\left(2 x \right)}