Riccati Equation. An equation of the form is called a

QUESTION:

An equation of the form

(18)     \(\frac{d y}{d x}=P(x) y^{2}+Q(x) y+R(x)\)

is called a generalized Riccati equation.

(a) If one solution—say, \(u(x)-\)of (18) is known, show that the substitution \(y=u+1/v\) reduces (18) to a linear equation in 𝑣.

(b) Given that \(u(x)=x\) is a solution to

\(\frac{d y}{d x}=x^{3}(y-x)^{2}+\frac{y}{x}\),

use the result of part (a) to find all the other solutions to this equation. (The particular solution \(u(x)=x\) can be found by inspection or by using a Taylor series method; see Section 8.1.)

Equation Transcription:

Text Transcription:

{dy}over{dx}=P(x)y^{2}+Q(x)y+R(x)

u(x)-

y=u+1/v

u(x)=x \\

{dy}over{dx}=x^{3}(y-x)^{2}+{y}over{x}

u(x)=x