1. **Stating the problem**: We want to find methods to solve first-order nonlinear differential equations. 2. **Separating variables**: If the equation can be written as $$\frac{dy}{dx} = g(x)h(y)$$, we separate variables to get $$\frac{1}{h(y)}dy = g(x)dx$$ and then integrate both sides. 3. **Substitution methods**: Using substitutions like $$v = y/x$$ or other transformations can simplify certain nonlinear equations into linear ones. 4. **Exact equations**: If the differential equation can be expressed as $$M(x,y) + N(x,y)\frac{dy}{dx} = 0$$ and satisfies $$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$, then it is exact and solvable using a potential function. 5. **Integrating factor method**: For some nonlinear equations not exact initially, multiplying by an integrating factor dependent on $$x$$ or $$y$$ can make them exact. 6. **Bernoulli's equation**: Equations in the form $$\frac{dy}{dx} + P(x)y = Q(x)y^n$$ with $$n \neq 0,1$$ can be transformed into linear equations using substitution $$v = y^{1-n}$$. 7. **Numerical methods**: When analytical solutions are difficult or impossible, methods like Euler's method, Runge-Kutta methods are used to approximate solutions. Each method depends on the form and properties of the differential equation, and often recognizing the type is the key first step.