Use the technique of Exercise to transform the Bernoulli equation into a linear equation....

Use the technique of Exercise to transform the Bernoulli equation into a linear equation. Find the general solution of the resulting linear equation.

Exercise

The presence of nonlinear terms prevents us from using the technique of this section. In special cases, a change of variable will transform the nonlinear equation into one that is linear. The equation known as Bernoulli’s equation,

was proposed for solution by James Bernoulli in December 1695. In 1696, Leibniz pointed out that the equation can be reduced to a linear equation by taking x1–n as the dependent variable. Show that the change of variable, z = x1–n, will transform the nonlinear Bernoulli equation into the linear equation

Hint: If z = x1–n, then

The equation

where ψ,ϕ, and χ are functions of t, is called the generalized Riccati equation. In general, the equation is not in tegrable by quadratures. However, suppose that one solution, say y = y1, is known.

(a) Show that the substitution y = y1 + z reduces the generalized Riccati equation to

which is an instance of Bernoulli’s equation (see Exercise 22).

(b) Use the fact that y1 = 1/t is a particular solution of

to find the equation’s general solution.