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Modelling with differential equations

Growth, decay, cooling, mixing: writing the equation from the sentence.

Every model starts with a sentence of the form “the rate of change of y is proportional to …”. Growth at 5% a year is y′ = 0.05y; cooling is T′ = −k(T − T_room); a mixing tank balances what flows in against what flows out. Write the sentence as an equation, solve, then fit the constant to a known value.

ਕੰਮ ਉਦਾਹਰਨ: y' = 0.05*y

Differential-Equations · Differential-Equations

Solve y' = 0.05*y

\frac{d}{d x} y{\left(x \right)} = 0.05 y{\left(x \right)}

ਕਦਮ ਦਰ ਕਦਮ

  1. \frac{d}{d x} y{\left(x \right)} = 0.05 y{\left(x \right)}

    The differential equation.

  2. \text{order } 1

    Order 1: the highest derivative present.

  3. \text{Separable}

    Move everything in y to one side with dy and everything in x to the other, then integrate both sides.

  4. 1.0 r - 0.05 = 0

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

  5. r = 0.05

    Its roots.

  6. y{\left(x \right)} = C_{1} e^{0.05 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)} = C_{1} e^{0.05 x}

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ਹੋਰ ਵਿੱਚ Differential Equations

Separable equationsFirst-order linear equationsSecond-order, constant coefficientsNonhomogeneous equations