# 2.6 | Substitutions and Transformations

If the right-hand side of the equation
$$\frac{dy}{dx} = f(x,y)$$
can be expressed as a function of the ratio $$y/x$$ alone, then we say the equation is homogeneous.
A first-order equation that can be written in the form
$$\frac{dy}{dx} + P(x)y = Q(x)y^n,$$
where $$P(x)$$ and $$Q(x)$$ are continuous on an interval $$(a,b)$$ and $$n$$ is a real number, is called a Bernoulli equation.
An equation that can be written in the form
$$(a_1 x + b_1 y + c_1) \; dx + (a_2 x + b_2 y + c_2) \; dy = 0,$$
where $$a_i$$’s, $$b_i$$’s, and $$c_i$$’s are constants, is called an equation with linear coefficients.