maths.free › Differential Equations › Modelling with differential equations

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.

Работен пример · y' = 0.05*y

Differential-Equations · Differential-Equations

Y' = 0.05*y

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

Стъпка по стъпка

  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.

Разкрийте отговора
y{\left(x \right)} = C_{1} e^{0.05 x}

Сега ти. Изберете проблем, или тип или нарисувате собствен. Всяка стъпка, снимка, отговор скрит, докато не попитате.

Запази си работата.

Безплатна сметка добавя бележки за всеки урок, запис на това, което сте завършили, решавате проблеми на едно място, и учител, който можете да попитате за тази страница. Самата математика е отворена за всеки, подписан или не.

Записване Вход

Символи, използвани тук

Докоснете всеки символ за пълното определение, снимка и какво означава всяка буква в него.

Как да се: Modelling with differential equations

  1. The differential equation.
  2. Order 1: the highest derivative present.
  3. Move everything in y to one side with dy and everything in x to the other, then integrate both sides.
  4. Characteristic equation of the homogeneous part (substitute y = e^{rx}).
  5. Its roots.
  6. General solution (C₁, C₂ … are arbitrary constants).
  7. Verified: substituting the solution back into the equation gives 0.

Въпроси, които хората задават

What is a differential equation?

An equation whose unknown is a function, relating it to its own derivatives. "The rate of growth is proportional to the population" is y′ = ky, and solving it means finding y as a function of time.

Why does the solution have arbitrary constants?

Integrating loses information: many functions share the same derivative. An n-th order equation has n constants, fixed by n initial or boundary conditions.

Още в Differential Equations

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