Exercise 27

Bernoulli's equation is an equation of the form $$y' = a(t) y + f(t) y^n,$$ where $n \neq 0$ or 1. Bernoulli's equation is nonlinear and cannot be solved by the techniques that we have used to solve first order linear equations.

1. Using the substitution $z = y^{1- n}$, show that we can transform Bernoulli's equation into the linear equation $$z' = (1 -n )a(t) z + (1-n) f(t).$$

2. Solve the equation $xy' + y = x^4y^3$.