55.10.13 problem 26

Internal problem ID [13420]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-2. Equations with cosine.
Problem number : 26
Date solved : Wednesday, October 01, 2025 at 11:30:01 AM
CAS classification : [_Riccati]

\begin{align*} \left (a \cos \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )-a \,\lambda ^{2} \cos \left (\lambda x \right )&=0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 204

ode:=(a*cos(lambda*x)+b)*(diff(y(x),x)-y(x)^2)-a*lambda^2*cos(lambda*x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (2 \,\operatorname {arctanh}\left (\frac {\tan \left (\frac {\lambda x}{2}\right ) \left (a -b \right )}{\sqrt {a^{2}-b^{2}}}\right ) \sqrt {a^{2}-b^{2}}\, a b \cos \left (\frac {\lambda x}{2}\right ) \sin \left (\frac {\lambda x}{2}\right )+2 \sqrt {a^{2}-b^{2}}\, c_1 a \cos \left (\frac {\lambda x}{2}\right ) \sin \left (\frac {\lambda x}{2}\right )+\left (a -b \right ) \left (a +b \right ) \left (a \cos \left (\frac {\lambda x}{2}\right )^{2}-\frac {a}{2}-\frac {b}{2}\right )\right ) \lambda }{\sqrt {a^{2}-b^{2}}\, \left (2 b \left (a \cos \left (\frac {\lambda x}{2}\right )^{2}-\frac {a}{2}+\frac {b}{2}\right ) \operatorname {arctanh}\left (\frac {\tan \left (\frac {\lambda x}{2}\right ) \left (a -b \right )}{\sqrt {a^{2}-b^{2}}}\right )-\sqrt {a^{2}-b^{2}}\, a \cos \left (\frac {\lambda x}{2}\right ) \sin \left (\frac {\lambda x}{2}\right )+2 \left (a \cos \left (\frac {\lambda x}{2}\right )^{2}-\frac {a}{2}+\frac {b}{2}\right ) c_1 \right )} \]

✓ Mathematica. Time used: 1.786 (sec). Leaf size: 231

ode=(a*Cos[\[Lambda]*x]+b)*(D[y[x],x]-y[x]^2)-a*\[Lambda]^2*Cos[\[Lambda]*x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^x-\frac {a \cos (\lambda K[1]) \lambda ^2+b y(x)^2+a \cos (\lambda K[1]) y(x)^2}{(b+a \cos (\lambda K[1])) (-a \lambda \sin (\lambda K[1])+b y(x)+a \cos (\lambda K[1]) y(x))^2}dK[1]+\int _1^{y(x)}\left (\frac {1}{(b K[2]+a \cos (x \lambda ) K[2]-a \lambda \sin (x \lambda ))^2}-\int _1^x\left (\frac {2 \left (a \cos (\lambda K[1]) \lambda ^2+b K[2]^2+a \cos (\lambda K[1]) K[2]^2\right )}{(b K[2]+a \cos (\lambda K[1]) K[2]-a \lambda \sin (\lambda K[1]))^3}-\frac {2 b K[2]+2 a \cos (\lambda K[1]) K[2]}{(b+a \cos (\lambda K[1])) (b K[2]+a \cos (\lambda K[1]) K[2]-a \lambda \sin (\lambda K[1]))^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]

✗ Sympy

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

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