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7.3 Bernoulli’s Equation

Bernoulli’s equation take the form

{dy\over dx} + p(x)y = q(x){y}^{n}\kern 1.66702pt .

In order to solve it, we convert it to linear type. Multiply both sides by {y}^{−n}(1 − n),

(1 − n){y}^{−n}{dy\over dx} + p(x)(1 − n){y}^{−n+1} = q(x)(1 − n)\kern 1.66702pt .

Now substitute z = {y}^{1−n}, using

{dz\over dx} = {dz\over dy} {dy\over dx} = (1 − n){y}^{−n}{dy\over dx}\kern 1.66702pt .

This leads to the equation

{dz\over dx} + (1 − n)p(x)z = (1 − n)q(x)\kern 1.66702pt .

If we then define \tilde{p}(x) = (1 − n)p(x) and \tilde{q}(x) = (1 − n)q(x), we have an equation of linear type, which can be dealt with through an itegrating factor.

Example 7.11: 

Solve {dy\over dx} + {1\over x}y = x{y}^{2}.

Solution: 

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