Question

A. Show that the solution (7) of the general linear equation (1) can be written in the form y cyi() +/2), (29) where c is an arbitrary constant.

b. Show that yi is a solution of the differential equation y + P()y 0. (30) corresponding to g(t) o
c. Show that y2 is a solution of the full linear equation (1). We see later (for example, in Section 3.5) that solutions of higher- order linear equations have a pattern similar to equation (29).
Bernoulli Equations. Sometimes it is possible to solve a nonlinear equation by making a change of the dependent variable that converts it into a linear equation. The most important such equation has the form
y'+ p(t)y=q(t)y^n

Answer 1

To show that yi is a solution of the differential equation y' + P(t)y = 0 corresponding to g(t), substitute yi into the differential equation and verify that it satisfies the equation.

Step-by-step explanation:

In order to show that yi is a solution of the differential equation y' + P(t)y = 0 corresponding to g(t), we need to substitute yi into the differential equation and verify that it satisfies the equation.

Let's start by differentiating yi with respect to t:

yi' = cyi(t)'

Now, substitute yi and yi' into the differential equation:

cyi(t)' + P(t)cyi(t) = 0

c * (yi(t)' + P(t)yi(t)) = 0

This equation is satisfied, which means that yi is indeed a solution of the differential equation y' + P(t)y = 0 corresponding to g(t).