Riccati Differential Equation

Author

Eric W. Weisstein
May 3, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialEquations/RiccatiDifferentialEquation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/RiccatiDifferentialEquation.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Boyce and DiPrima

FullSimplify[w'[z]==P[z]+Q[z]w[z]+R[z]w[z]^2/.w->Function[z,-y'[z]/y[z]/R[z]]]

DSolve[P[z] ((R[z])^2) y[z]-(Q[z] R[z]+R'[z]) y'[z]+R[z] y''[z]==0,y'[z],z]

Other Forms

y'(z)==R(z) ((y(z))^2)+Q(z) y(z)+P(z)

v4.2

DSolve[y'[z]==P[z]+Q[z]y[z]+R[z]y[z]^2,y[z],z]

v5.0

DSolve[y'[z]==P[z]+Q[z]y[z]+R[z]y[z]^2,y[z],z]

w''(z) (z^2)+((z^2)-n (n+1)) w(z)==0

DSolve[z^2 w''[z]+(z^2-n(n+1))w[z]==0,w[z],z]

FullSimplify[DSolve[y'[z]==a z^n+b y[z]^2,y[z],z]]

Table[% /.n->-4m/(2m+1),{m,0,3}]

Table[%4 /.n->-4m/(2m-1),{m,0,3}]