Solution Found!
as Riccati’s equation.(a) A Riccati equation can be solved
Chapter 2, Problem 35E(choose chapter or problem)
The differential equation \(d y / d x=P(x)+Q(x) y+R(x) y^{2}\) is known as Riccati’s equation.
(a) A Riccati equation can be solved by a succession of two substitutions provided that we know a particular solution \(y_{1}\) of the equation. Show that the substitution \(y=y_{1}+u\) reduces Riccati’s equation to a Bernoulli equation (4) with n = 2. The Bernoulli equation can then be reduced to a linear equation by the substitution \(w=u^{-1}\) .
(b) Find a one-parameter family of solutions for the differential equation
\(\frac{d y}{d x}=-\frac{4}{x^{2}}-\frac{1}{x} y+y^{2}\)
where \(y_{1}=2 / x\) is a known solution of the equation.
Text Transcription:
dy/dx = P(x) + Q(x)y + R(x)y^2
y_1
y = y_1 + u
w=u^-1
dy/dx = -4/x^2 - 1/x y + y^2
y_1 = 2/x
Questions & Answers
QUESTION:
The differential equation \(d y / d x=P(x)+Q(x) y+R(x) y^{2}\) is known as Riccati’s equation.
(a) A Riccati equation can be solved by a succession of two substitutions provided that we know a particular solution \(y_{1}\) of the equation. Show that the substitution \(y=y_{1}+u\) reduces Riccati’s equation to a Bernoulli equation (4) with n = 2. The Bernoulli equation can then be reduced to a linear equation by the substitution \(w=u^{-1}\) .
(b) Find a one-parameter family of solutions for the differential equation
\(\frac{d y}{d x}=-\frac{4}{x^{2}}-\frac{1}{x} y+y^{2}\)
where \(y_{1}=2 / x\) is a known solution of the equation.
Text Transcription:
dy/dx = P(x) + Q(x)y + R(x)y^2
y_1
y = y_1 + u
w=u^-1
dy/dx = -4/x^2 - 1/x y + y^2
y_1 = 2/x
ANSWER:Step 1 of 3
The differential equation 