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.

Exemplo resolvido · 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)}

Passo a passo

  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.

Revelar a resposta
y{\left(x \right)} = C_{1} e^{0.05 x}

Agora tu Escolha um problema, ou digite ou desenhe o seu próprio. Cada passo, uma imagem, a resposta escondida até que você pergunte.

Guarde o seu próprio trabalho

Uma conta gratuita adiciona notas sobre cada lição, um registro do que você terminou, seus problemas resolvidos em um lugar, e um tutor que você pode perguntar sobre esta página. A própria matemática está aberta a todos, assinada ou não.

Inscrever-se Login

Símbolos usados aqui

Toque em qualquer símbolo para a definição completa, uma imagem e o que cada letra nela significa.

Como: 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.

Perguntas que as pessoas fazem

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.

Mais em Differential Equations

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