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.

E 2.6 Exercises