61.10.7 problem 20
Internal
problem
ID
[12194]
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
:
20
Date
solved
:
Tuesday, January 28, 2025 at 01:01:50 AM
CAS
classification
:
[_Riccati]
\begin{align*} 2 y^{\prime }&=\left (\lambda +a -a \cos \left (\lambda x \right )\right ) y^{2}+\lambda -a -a \cos \left (\lambda x \right ) \end{align*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 120
dsolve(2*diff(y(x),x)=(lambda+a-a*cos(lambda*x))*y(x)^2+lambda-a-a*cos(lambda*x),y(x), singsol=all)
\[
y = \frac {\cot \left (\frac {\lambda x}{2}\right ) \lambda \left (\int \operatorname {csgn}\left (\sin \left (\frac {\lambda x}{2}\right )\right ) \left (\csc \left (\frac {\lambda x}{2}\right )^{2} \lambda +2 a \right ) {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }}d x \right ) c_{1} +2 \csc \left (\frac {\lambda x}{2}\right )^{2} \operatorname {csgn}\left (\sin \left (\frac {\lambda x}{2}\right )\right ) {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} c_{1} \lambda -2 i \cot \left (\frac {\lambda x}{2}\right )}{-\lambda \left (\int \operatorname {csgn}\left (\sin \left (\frac {\lambda x}{2}\right )\right ) \left (\csc \left (\frac {\lambda x}{2}\right )^{2} \lambda +2 a \right ) {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }}d x \right ) c_{1} +2 i}
\]
✓ Solution by Mathematica
Time used: 10.395 (sec). Leaf size: 321
DSolve[2*D[y[x],x]==(\[Lambda]+a-a*Cos[\[Lambda]*x])*y[x]^2+\[Lambda]-a-a*Cos[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {2 \left (c_1 \cot \left (\frac {\lambda x}{2}\right ) \int _1^xe^{-\frac {2 a \sin ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} \left (\lambda \csc ^2\left (\frac {1}{2} \lambda K[1]\right )+2 a\right )dK[1]+2 c_1 \csc ^2\left (\frac {\lambda x}{2}\right ) e^{-\frac {2 a \sin ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}+\cot \left (\frac {\lambda x}{2}\right )\right )}{2+2 c_1 \int _1^xe^{-\frac {2 a \sin ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} \left (\lambda \csc ^2\left (\frac {1}{2} \lambda K[1]\right )+2 a\right )dK[1]} \\
y(x)\to \frac {1}{2} \csc ^2\left (\frac {\lambda x}{2}\right ) \left (-\frac {4 e^{-\frac {2 a \sin ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}}{\int _1^xe^{-\frac {2 a \sin ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} \left (\lambda \csc ^2\left (\frac {1}{2} \lambda K[1]\right )+2 a\right )dK[1]}-\sin (\lambda x)\right ) \\
y(x)\to \frac {1}{2} \csc ^2\left (\frac {\lambda x}{2}\right ) \left (-\frac {4 e^{-\frac {2 a \sin ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}}{\int _1^xe^{-\frac {2 a \sin ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} \left (\lambda \csc ^2\left (\frac {1}{2} \lambda K[1]\right )+2 a\right )dK[1]}-\sin (\lambda x)\right ) \\
\end{align*}