The Bernoulli equation has the form

+ f(x)y = g(x)yk

(1)

where f and g are continuous real functions and k is a constant (≠0, ≠1). Such an equation is got e.g. in examining the motion of a body when the resistance of medium depends on the velocity v as

The real function y can be solved from (1) explicitly. To do this, divide first both sides by y^k. It yields

y−k + f(x)y−k+1 = g(x).

(2)

The substitution

z := y−k+1

(3)

transforms (2) into

which is a linear differential equation of first order. When one has obtained its general solution and made in this the substitution (3), then one has solved the Bernoulli equation (1).

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