- First order differential equations involve only the first derivative of the unknown function
- Variable separable equations involve separating the variables in the equation and integrating each side to obtain the solution
- Homogeneous equations involve replacing the function with a new variable that simplifies the equation, and then solving it using separation of variables
- Exact equations involve finding an integrating factor that makes the equation exact, and then integrating to obtain the solution
- Bernoulli's equation involves transforming a nonlinear first order equation into a linear equation using a substitution, and then solving it using an integrating factor
Important Formulas to Remember in these topic
Explanation and solution methods for first order differential equations, including:
- Variable separable equations, such as:
$$\frac{dy}{dx} = f(x)g(y) \Rightarrow \int\frac{1}{g(y)}dy = \int f(x)dx + C$$
- Homogeneous equations, such as:
$$\frac{dy}{dx} = f\left(\frac{y}{x}\right) \Rightarrow y = vx \Rightarrow \frac{dv}{dx} = \frac{v-f(v)}{x}$$
- Exact equations, such as:
$$M(x,y)dx + N(x,y)dy = 0 \Rightarrow \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \Rightarrow \text{solution } F(x,y) = C$$
- Linear differential equation of the form dy/dx+Py=Q, such as:
$$\frac{dy}{dx} + P(x)y = Q(x) \Rightarrow y = e^{-\int P(x)dx}\left(\int Q(x)e^{\int P(x)dx}dx + C\right)$$
- Bernoulli's equation, such as:
$$\frac{dy}{dx} + P(x)y = Q(x)y^n \Rightarrow z = y^{1-n} \Rightarrow \frac{dz}{dx} + (1-n)P(x)z = (1-n)Q(x)$$