Bernoulli First-Order ODE

A Bernoulli first-order ode has the form

where g(t) and h(t) are given functions and n does not equal 1.

Solution Procedure

The idea is to convert the Bernoulli equation into a linear ode. We make the substitution

Differentiating this expression we have

Solving for y'(t), we have

Substituting this expression into the original ode (*), we have

Dividing both sides by y^n/(1-n) we have

Replacing y^(1-n) by v, we obtain

This is a linear ode in v(t)! Solve for v(t). We obtain y(t) using the formula

Example

Consider the ode

This is a Bernoulli equation with n=3, g(t)=5, h(t)=-5t. We make the substitution

Applying the chain rule, we have

Solving for y'(t), we have

Substituting for y'(t) in the differential equation we have

Dividing both sides by -.5y^3, we have

Note that 1/y^2=v. Hence, we have

This is a linear ode for the dependent variable v(t). The solution is

where C is a constant. Substiting v=1/y^2, we have

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