List of basic formulas and types of first-order differential equations:

  1. General first order equation:
    [
    \frac{dy}{dx} = f(x, y)
    ]
  2. Separable variables:
    [
    \frac{dy}{dx} = g(y)h(x) \quad \Rightarrow \quad \frac{1}{g(y)} dy = h(x) dx
    ]
  3. Linear Equations:
    [
    \frac{dy}{dx} + P(x)y = Q(x)
    ]
    Solution:
    [
    y(x) = e^{-\int P(x)dx} \left( \int Q (x)e^{\int P(x)dx}dx + C \right)
    ]
  4. Bernoulli’s equations:
    [
    \frac{dy}{dx} + P(x)y = Q(x)y^n
    ]
    (where ( n \neq 0, 1 )). Transformed into a linear equation.
  5. Equations with constant coefficients:
    [
    \frac{dy}{dx} + ay = b
    ]
    Solution:
    [
    y(x) = Ce^{-ax} + \frac{b}{a}
    ]
  6. Equations with homogeneous functions:
    If ( f(tx, ty) = t^nf(x, y) ), then we can use the substitution ( y = vx ).
  7. Equations with full derivatives:
    If ( M(x, y)dx + N(x, y)dy = 0 ) and ( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} ).

These are the basic types and formulas for first order differential equations.

От Math

Добавить комментарий

Ваш адрес email не будет опубликован. Обязательные поля помечены *