61.10.13 problem 26
Internal
problem
ID
[12200]
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
:
Tuesday, January 28, 2025 at 01:06:54 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*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 204
dsolve((a*cos(lambda*x)+b)*(diff(y(x),x)-y(x)^2)-a*lambda^2*cos(lambda*x)=0,y(x), singsol=all)
\[
y = \frac {\left (2 \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {\lambda x}{2}\right )}{\sqrt {a^{2}-b^{2}}}\right ) \sqrt {a^{2}-b^{2}}\, a b \cos \left (\frac {\lambda x}{2}\right ) \sin \left (\frac {\lambda x}{2}\right )-2 \sqrt {a^{2}-b^{2}}\, c_{1} a \cos \left (\frac {\lambda x}{2}\right ) \sin \left (\frac {\lambda x}{2}\right )+\left (a +b \right ) \left (a -b \right ) \left (a \cos \left (\frac {\lambda x}{2}\right )^{2}-\frac {a}{2}-\frac {b}{2}\right )\right ) \lambda }{\sqrt {a^{2}-b^{2}}\, \left (2 \left (a \cos \left (\frac {\lambda x}{2}\right )^{2}-\frac {a}{2}+\frac {b}{2}\right ) b \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {\lambda x}{2}\right )}{\sqrt {a^{2}-b^{2}}}\right )-a \sqrt {a^{2}-b^{2}}\, \cos \left (\frac {\lambda x}{2}\right ) \sin \left (\frac {\lambda x}{2}\right )-2 c_{1} \left (a \cos \left (\frac {\lambda x}{2}\right )^{2}-\frac {a}{2}+\frac {b}{2}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 2.702 (sec). Leaf size: 231
DSolve[(a*Cos[\[Lambda]*x]+b)*(D[y[x],x]-y[x]^2)-a*\[Lambda]^2*Cos[\[Lambda]*x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^x-\frac {a \cos (\lambda K[1]) \lambda ^2+b y(x)^2+a \cos (\lambda K[1]) y(x)^2}{(b+a \cos (\lambda K[1])) (-a \lambda \sin (\lambda K[1])+b y(x)+a \cos (\lambda K[1]) y(x))^2}dK[1]+\int _1^{y(x)}\left (\frac {1}{(b K[2]+a \cos (x \lambda ) K[2]-a \lambda \sin (x \lambda ))^2}-\int _1^x\left (\frac {2 \left (a \cos (\lambda K[1]) \lambda ^2+b K[2]^2+a \cos (\lambda K[1]) K[2]^2\right )}{(b K[2]+a \cos (\lambda K[1]) K[2]-a \lambda \sin (\lambda K[1]))^3}-\frac {2 b K[2]+2 a \cos (\lambda K[1]) K[2]}{(b+a \cos (\lambda K[1])) (b K[2]+a \cos (\lambda K[1]) K[2]-a \lambda \sin (\lambda K[1]))^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ]
\]