List of basic formulas and types of first-order differential equations:
- General first order equation:
[
\frac{dy}{dx} = f(x, y)
] - Separable variables:
[
\frac{dy}{dx} = g(y)h(x) \quad \Rightarrow \quad \frac{1}{g(y)} dy = h(x) dx
] - 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)
] - Bernoulli’s equations:
[
\frac{dy}{dx} + P(x)y = Q(x)y^n
]
(where ( n \neq 0, 1 )). Transformed into a linear equation. - Equations with constant coefficients:
[
\frac{dy}{dx} + ay = b
]
Solution:
[
y(x) = Ce^{-ax} + \frac{b}{a}
] - Equations with homogeneous functions:
If ( f(tx, ty) = t^nf(x, y) ), then we can use the substitution ( y = vx ). - 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.