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Second-order, constant coefficients

ay″ + by′ + cy = 0 and the characteristic equation: oscillation, damping, growth.

Try y = e^{rx} and the equation becomes a quadratic in r — the characteristic equation. Real distinct roots give exponentials, a repeated root adds an x, complex roots give sines and cosines. y″ + y = 0 is the simple harmonic oscillator: every pendulum, spring and LC circuit.

ਕੰਮ ਉਦਾਹਰਨ: y'' + y = 0

Differential-Equations · Differential-Equations

Solve y'' + y = 0

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

ਕਦਮ ਦਰ ਕਦਮ

  1. 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} + 1 = 0

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

  5. r = - i,\ r = i

    Its roots.

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

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

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ਹੋਰ ਵਿੱਚ Differential Equations

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