2.10.4 Problem 19
Internal
problem
ID
[13381]
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
:
19
Date
solved
:
Sunday, January 18, 2026 at 07:42:07 PM
CAS
classification
:
[_Riccati]
2.10.4.1 Solved using first_order_ode_riccati
5.338 (sec)
Entering first order ode riccati solver
\begin{align*}
y^{\prime }&=\lambda \cos \left (\lambda x \right ) y^{2}+\lambda \cos \left (\lambda x \right )^{3} \\
\end{align*}
In canonical form the ODE is \begin{align*} y' &= F(x,y)\\ &= \lambda \cos \left (\lambda x \right ) y^{2}+\lambda \cos \left (\lambda x \right )^{3} \end{align*}
This is a Riccati ODE. Comparing the ODE to solve
\[
y' = \lambda \cos \left (\lambda x \right ) y^{2}+\lambda \cos \left (\lambda x \right )^{3}
\]
With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]
Shows
that \(f_0(x)=\lambda \cos \left (\lambda x \right )^{3}\), \(f_1(x)=0\) and \(f_2(x)=\lambda \cos \left (\lambda x \right )\). Let \begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{u \lambda \cos \left (\lambda x \right )} \tag {1} \end{align*}
Using the above substitution in the given ODE results (after some simplification) in a second
order ODE to solve for \(u(x)\) which is
\begin{align*} f_2 u''(x) -\left ( f_2' + f_1 f_2 \right ) u'(x) + f_2^2 f_0 u(x) &= 0 \tag {2} \end{align*}
But
\begin{align*} f_2' &=-\sin \left (\lambda x \right ) \lambda ^{2}\\ f_1 f_2 &=0\\ f_2^2 f_0 &=\lambda ^{3} \cos \left (\lambda x \right )^{5} \end{align*}
Substituting the above terms back in equation (2) gives
\[
\lambda \cos \left (\lambda x \right ) u^{\prime \prime }\left (x \right )+\sin \left (\lambda x \right ) \lambda ^{2} u^{\prime }\left (x \right )+\lambda ^{3} \cos \left (\lambda x \right )^{5} u \left (x \right ) = 0
\]
Unable to solve. Will ask Maple to solve
this ode now.
The solution for \(u \left (x \right )\) is
\begin{equation}
\tag{3} u \left (x \right ) = c_1 \sin \left (\lambda x \right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \operatorname {KummerM}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )+c_2 \sin \left (\lambda x \right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \operatorname {KummerU}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )
\end{equation}
Taking derivative gives \begin{equation}
\tag{4} u^{\prime }\left (x \right ) = c_1 \lambda \cos \left (\lambda x \right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \operatorname {KummerM}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )+c_1 \sin \left (\lambda x \right )^{2} \lambda \cos \left (\lambda x \right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \operatorname {KummerM}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )+2 c_1 \,{\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \left (\left (-\sin \left (\lambda x \right )^{2}-\frac {1}{2}\right ) \operatorname {KummerM}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )+\frac {1}{2}\right ) \cos \left (\lambda x \right ) \lambda +c_2 \lambda \cos \left (\lambda x \right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \operatorname {KummerU}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )+c_2 \sin \left (\lambda x \right )^{2} \lambda \cos \left (\lambda x \right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \operatorname {KummerU}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )+2 c_2 \,{\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \left (\left (-\sin \left (\lambda x \right )^{2}-\frac {1}{2}\right ) \operatorname {KummerU}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )-1\right ) \cos \left (\lambda x \right ) \lambda
\end{equation}
Substituting equations (3,4) into (1) results in \begin{align*}
y &= \frac {-u'}{f_2 u} \\
y &= \frac {-u'}{u \lambda \cos \left (\lambda x \right )} \\
y &= -\frac {c_1 \lambda \cos \left (\lambda x \right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \operatorname {KummerM}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )+c_1 \sin \left (\lambda x \right )^{2} \lambda \cos \left (\lambda x \right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \operatorname {KummerM}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )+2 c_1 \,{\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \left (\left (-\sin \left (\lambda x \right )^{2}-\frac {1}{2}\right ) \operatorname {KummerM}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )+\frac {1}{2}\right ) \cos \left (\lambda x \right ) \lambda +c_2 \lambda \cos \left (\lambda x \right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \operatorname {KummerU}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )+c_2 \sin \left (\lambda x \right )^{2} \lambda \cos \left (\lambda x \right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \operatorname {KummerU}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )+2 c_2 \,{\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \left (\left (-\sin \left (\lambda x \right )^{2}-\frac {1}{2}\right ) \operatorname {KummerU}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )-1\right ) \cos \left (\lambda x \right ) \lambda }{\lambda \cos \left (\lambda x \right ) \left (c_1 \sin \left (\lambda x \right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \operatorname {KummerM}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )+c_2 \sin \left (\lambda x \right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \operatorname {KummerU}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )\right )} \\
\end{align*}
Doing change of constants, the above solution becomes \[
y = -\frac {\lambda \cos \left (\lambda x \right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \operatorname {KummerM}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )+\sin \left (\lambda x \right )^{2} \lambda \cos \left (\lambda x \right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \operatorname {KummerM}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )+2 \,{\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \left (\left (-\sin \left (\lambda x \right )^{2}-\frac {1}{2}\right ) \operatorname {KummerM}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )+\frac {1}{2}\right ) \cos \left (\lambda x \right ) \lambda +c_3 \lambda \cos \left (\lambda x \right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \operatorname {KummerU}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )+c_3 \sin \left (\lambda x \right )^{2} \lambda \cos \left (\lambda x \right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \operatorname {KummerU}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )+2 c_3 \,{\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \left (\left (-\sin \left (\lambda x \right )^{2}-\frac {1}{2}\right ) \operatorname {KummerU}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )-1\right ) \cos \left (\lambda x \right ) \lambda }{\lambda \cos \left (\lambda x \right ) \left (\sin \left (\lambda x \right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \operatorname {KummerM}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )+c_3 \sin \left (\lambda x \right ) {\mathrm e}^{-\frac {\cos \left (\lambda x \right )^{2}}{2}} \operatorname {KummerU}\left (1, \frac {3}{2}, -\sin \left (\lambda x \right )^{2}\right )\right )}
\]
Simplifying the above gives
\begin{align*}
y &= -\frac {4 \csc \left (\lambda x \right ) \left (-\frac {\left (-\frac {1}{2}+c_3 \right ) \sin \left (\lambda x \right )^{2} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\sin \left (\lambda x \right )^{2}}\right )}{2}+\left (-\frac {1}{2}+c_3 \right ) {\mathrm e}^{\sin \left (\lambda x \right )^{2}} \sqrt {-\sin \left (\lambda x \right )^{2}}+\frac {\sin \left (\lambda x \right )^{2} \sqrt {\pi }\, c_3}{2}\right )}{\sqrt {\pi }\, \left (\operatorname {erf}\left (\sqrt {-\sin \left (\lambda x \right )^{2}}\right ) \left (2 c_3 -1\right )-2 c_3 \right )} \\
\end{align*}
Summary of solutions found
\begin{align*}
y &= -\frac {4 \csc \left (\lambda x \right ) \left (-\frac {\left (-\frac {1}{2}+c_3 \right ) \sin \left (\lambda x \right )^{2} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\sin \left (\lambda x \right )^{2}}\right )}{2}+\left (-\frac {1}{2}+c_3 \right ) {\mathrm e}^{\sin \left (\lambda x \right )^{2}} \sqrt {-\sin \left (\lambda x \right )^{2}}+\frac {\sin \left (\lambda x \right )^{2} \sqrt {\pi }\, c_3}{2}\right )}{\sqrt {\pi }\, \left (\operatorname {erf}\left (\sqrt {-\sin \left (\lambda x \right )^{2}}\right ) \left (2 c_3 -1\right )-2 c_3 \right )} \\
\end{align*}
2.10.4.2 ✓ Maple. Time used: 0.003 (sec). Leaf size: 95
ode:=diff(y(x),x) = lambda*cos(lambda*x)*y(x)^2+lambda*cos(lambda*x)^3;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {4 \csc \left (\lambda x \right ) \left (-\frac {\left (c_1 -\frac {1}{2}\right ) \sin \left (\lambda x \right )^{2} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-\sin \left (\lambda x \right )^{2}}\right )}{2}+\left (c_1 -\frac {1}{2}\right ) {\mathrm e}^{\sin \left (\lambda x \right )^{2}} \sqrt {-\sin \left (\lambda x \right )^{2}}+\frac {\sin \left (\lambda x \right )^{2} \sqrt {\pi }\, c_1}{2}\right )}{\sqrt {\pi }\, \left (\operatorname {erf}\left (\sqrt {-\sin \left (\lambda x \right )^{2}}\right ) \left (2 c_1 -1\right )-2 c_1 \right )}
\]
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 Riccati sub-methods:
trying Riccati_symmetries
trying Riccati to 2nd Order
-> Calling odsolve with the ODE, diff(diff(y(x),x),x) = -lambda*sin(lambda*x
)/cos(lambda*x)*diff(y(x),x)-cos(lambda*x)^4*lambda^2*y(x), y(x)
*** Sublevel 2 ***
Methods for second order ODEs:
--- Trying classification methods ---
trying a symmetry of the form [xi=0, eta=F(x)]
checking if the LODE is missing y
-> Heun: Equivalence to the GHE or one of its 4 confluent cases under a \
power @ Moebius
-> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int\
(r(x), dx)) * 2F1([a1, a2], [b1], f)
-> Trying changes of variables to rationalize or make the ODE simpler
trying a quadrature
checking if the LODE has constant coefficients
checking if the LODE is of Euler type
trying a symmetry of the form [xi=0, eta=F(x)]
checking if the LODE is missing y
-> Trying a Liouvillian solution using Kovacics algorithm
A Liouvillian solution exists
Reducible group (found an exponential solution)
Group is reducible, not completely reducible
Solution has integrals. Trying a special function solution free of \
integrals...
-> Trying a solution in terms of special functions:
-> Bessel
-> elliptic
-> Legendre
-> Kummer
-> hyper3: Equivalence to 1F1 under a power @ Moebius
<- hyper3 successful: received ODE is equivalent to the 1F1 O\
DE
<- Kummer successful
<- special function solution successful
-> Trying to convert hypergeometric functions to elementary form\
...
<- elementary form is not straightforward to achieve - returning\
special function solution free of uncomputed integrals
<- Kovacics algorithm successful
Change of variables used:
[x = arccos(t)/lambda]
Linear ODE actually solved:
16*t^5*u(t)-16*diff(u(t),t)+(-16*t^3+16*t)*diff(diff(u(t),t),t) = 0
<- change of variables successful
<- Riccati to 2nd Order successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\lambda \cos \left (\lambda x \right ) y \left (x \right )^{2}+\lambda \cos \left (\lambda x \right )^{3} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\lambda \cos \left (\lambda x \right ) y \left (x \right )^{2}+\lambda \cos \left (\lambda x \right )^{3} \end {array} \]
2.10.4.3 ✗ Mathematica
ode=D[y[x],x]==\[Lambda]*Cos[\[Lambda]*x]*y[x]^2+\[Lambda]*Cos[\[Lambda]*x]^3;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.10.4.4 ✗ Sympy
from sympy import *
x = symbols("x")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(-lambda_*y(x)**2*cos(lambda_*x) - lambda_*cos(lambda_*x)**3 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -lambda_*(y(x)**2 + cos(lambda_*x)**2)*cos(lambda_*x) + Derivati
Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0]
Sympy version 1.14.0
classify_ode(ode,func=y(x))
('factorable', '1st_power_series', 'lie_group')