2.19.29 Problem 30

Internal problem ID [13477]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-1. Equations containing arbitrary functions (but not containing their derivatives).
Problem number : 30
Date solved : Friday, December 19, 2025 at 04:49:57 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2} f \left (x \right )-f \left (x \right ) a^{2}+a \lambda \sin \left (\lambda x \right )+a^{2} f \left (x \right ) \sin \left (\lambda x \right )^{2} \\ \end{align*}

Unknown ode type.

2.19.29.1 ✗ Maple

ode:=diff(y(x),x) = f(x)*y(x)^2-a^2*f(x)+a*lambda*sin(lambda*x)+a^2*f(x)*sin(lambda*x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
trying inverse linear 
trying homogeneous types: 
trying Chini 
differential order: 1; looking for linear symmetries 
trying exact 
Looking for potential symmetries 
trying Riccati 
trying inverse_Riccati 
trying 1st order ODE linearizable_by_differentiation 
--- Trying Lie symmetry methods, 1st order --- 
   -> Computing symmetries using: way = 4 
   -> Computing symmetries using: way = 2 
   -> Computing symmetries using: way = 6 
trying symmetry patterns for 1st order ODEs 
-> trying a symmetry pattern of the form [F(x)*G(y), 0] 
-> trying a symmetry pattern of the form [0, F(x)*G(y)] 
-> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] 
   -> Computing symmetries using: way = HINT 
   -> Calling odsolve with the ODE, diff(y(x),x) = 0, y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x) = -4*y(x)*x/(a^2-2*x^2), y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x)+y(x)*diff(f(x),x)/f(x), y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x)-y(x)*(2*f(x)*sin(2*lambda*x)*a 
*lambda-diff(f(x),x)*cos(2*lambda*x)*a+2*cos(lambda*x)*lambda^2-a*diff(f(x),x)) 
/(f(x)*cos(2*lambda*x)*a+f(x)*a-2*lambda*sin(lambda*x)), y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x) = -y(x)*diff(f(x),x)/f(x), y(x 
) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x), y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x)+4*y(x)*x/(a^2-2*x^2), y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
-> trying a symmetry pattern of the form [F(x),G(x)] 
-> trying a symmetry pattern of the form [F(y),G(y)] 
-> trying a symmetry pattern of the form [F(x)+G(y), 0] 
-> trying a symmetry pattern of the form [0, F(x)+G(y)] 
-> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
-> trying a symmetry pattern of conformal type
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=f \left (x \right ) y \left (x \right )^{2}-a^{2} f \left (x \right )+a \lambda \sin \left (\lambda x \right )+a^{2} f \left (x \right ) \sin \left (\lambda x \right )^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=f \left (x \right ) y \left (x \right )^{2}-a^{2} f \left (x \right )+a \lambda \sin \left (\lambda x \right )+a^{2} f \left (x \right ) \sin \left (\lambda x \right )^{2} \end {array} \]

2.19.29.2 ✗ Mathematica

ode=D[y[x],x]==f[x]*y[x]^2-a^2*f[x]+a*\[Lambda]*Sin[\[Lambda]*x]+a^2*f[x]*Sin[\[Lambda]*x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.19.29.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
lambda_ = symbols("lambda_") 
y = Function("y") 
f = Function("f") 
ode = Eq(-a**2*f(x)*sin(lambda_*x)**2 + a**2*f(x) - a*lambda_*sin(lambda_*x) - f(x)*y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE a**2*f(x)*cos(lambda_*x)**2 - a*lambda_*sin(lambda_*x) - f(x)*y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method