61.9.13 problem 13
Internal
problem
ID
[12108]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.6-1.
Equations
with
sine
Problem
number
:
13
Date
solved
:
Wednesday, March 05, 2025 at 04:26:46 PM
CAS
classification
:
[_Riccati]
\begin{align*} \left (a \sin \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )-a \,\lambda ^{2} \sin \left (\lambda x \right )&=0 \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 261
ode:=(a*sin(lambda*x)+b)*(diff(y(x),x)-y(x)^2)-a*lambda^2*sin(lambda*x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\lambda \left (-2 a \left (\cos \left (\frac {\lambda x}{2}\right )^{2}-\frac {1}{2}\right ) \left (a +b \right ) \sqrt {-a^{2}+b^{2}}\, \left (a -b \right ) b^{2} \arctan \left (\frac {b \tan \left (\frac {\lambda x}{2}\right )+a}{\sqrt {-a^{2}+b^{2}}}\right )-2 c_{1} a \left (\cos \left (\frac {\lambda x}{2}\right )^{2}-\frac {1}{2}\right ) \sqrt {-a^{2}+b^{2}}+\left (a^{2} \cos \left (\frac {\lambda x}{2}\right )^{2}-\sin \left (\frac {\lambda x}{2}\right ) a b \cos \left (\frac {\lambda x}{2}\right )-\frac {a^{2}}{2}+\frac {b^{2}}{2}\right ) \left (a +b \right )^{2} \left (a -b \right )^{2}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (2 \left (\sin \left (\frac {\lambda x}{2}\right ) a \cos \left (\frac {\lambda x}{2}\right )+\frac {b}{2}\right ) \left (a +b \right ) \left (a -b \right ) b^{2} \arctan \left (\frac {b \tan \left (\frac {\lambda x}{2}\right )+a}{\sqrt {-a^{2}+b^{2}}}\right )+a \cos \left (\frac {\lambda x}{2}\right ) \left (a -b \right ) \left (a +b \right ) \left (a \sin \left (\frac {\lambda x}{2}\right )+b \cos \left (\frac {\lambda x}{2}\right )\right ) \sqrt {-a^{2}+b^{2}}+2 c_{1} \left (\sin \left (\frac {\lambda x}{2}\right ) a \cos \left (\frac {\lambda x}{2}\right )+\frac {b}{2}\right )\right )}
\]
✓ Mathematica. Time used: 11.973 (sec). Leaf size: 227
ode=(a*Sin[\[Lambda]*x]+b)*(D[y[x],x]-y[x]^2)-a*\[Lambda]^2*Sin[\[Lambda]*x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^x-\frac {a \sin (\lambda K[1]) \lambda ^2+b y(x)^2+a \sin (\lambda K[1]) y(x)^2}{(b+a \sin (\lambda K[1])) (a \lambda \cos (\lambda K[1])+b y(x)+a \sin (\lambda K[1]) y(x))^2}dK[1]+\int _1^{y(x)}\left (\frac {1}{(a \lambda \cos (x \lambda )+b K[2]+a K[2] \sin (x \lambda ))^2}-\int _1^x\left (\frac {2 \left (a \sin (\lambda K[1]) \lambda ^2+b K[2]^2+a K[2]^2 \sin (\lambda K[1])\right )}{(a \lambda \cos (\lambda K[1])+b K[2]+a K[2] \sin (\lambda K[1]))^3}-\frac {2 b K[2]+2 a \sin (\lambda K[1]) K[2]}{(b+a \sin (\lambda K[1])) (a \lambda \cos (\lambda K[1])+b K[2]+a K[2] \sin (\lambda K[1]))^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
cg = symbols("cg")
y = Function("y")
ode = Eq(-a*cg**2*sin(cg*x) + (a*sin(cg*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*cg**2*sin(cg*x) + a*y(x)**2*sin(cg*x) + b*y(x)**2)/(a*sin(cg*x) + b) cannot be solved by the factorable group method