61.10.12 problem 25
Internal
problem
ID
[12199]
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
:
25
Date
solved
:
Tuesday, January 28, 2025 at 07:28:18 PM
CAS
classification
:
[_Riccati]
\begin{align*} \left (a \cos \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+c \cos \left (\mu x \right ) y-d^{2}+c d \cos \left (\mu x \right ) \end{align*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 268
dsolve((a*cos(lambda*x)+b)*diff(y(x),x)=y(x)^2+c*cos(mu*x)*y(x)-d^2+c*d*cos(mu*x),y(x), singsol=all)
\[
y = \frac {-d \left (\int \frac {{\mathrm e}^{\frac {c \left (\int \frac {\cos \left (\mu x \right )}{a \cos \left (\lambda x \right )+b}d x \right ) \sqrt {a^{2}-b^{2}}\, \lambda -4 d \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {\lambda x}{2}\right )}{\sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \lambda }}}{a \cos \left (\lambda x \right )+b}d x \right )+d c_{1} -{\mathrm e}^{\frac {c \left (\int \frac {\cos \left (\mu x \right )}{a \cos \left (\lambda x \right )+b}d x \right ) \sqrt {a^{2}-b^{2}}\, \lambda -4 d \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {\lambda x}{2}\right )}{\sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \lambda }}}{\int \frac {{\mathrm e}^{\frac {c \left (\int \frac {\cos \left (\mu x \right )}{a \cos \left (\lambda x \right )+b}d x \right ) \sqrt {a^{2}-b^{2}}\, \lambda -4 d \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {\lambda x}{2}\right )}{\sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \lambda }}}{a \cos \left (\lambda x \right )+b}d x -c_{1}}
\]
✓ Solution by Mathematica
Time used: 3.763 (sec). Leaf size: 289
DSolve[(a*Cos[\[Lambda]*x]+b)*D[y[x],x]==y[x]^2+c*Cos[\[Mu]*x]*y[x]-d^2+c*d*Cos[\[Mu]*x],y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {2 d-c \cos (\mu K[1])}{b+a \cos (\lambda K[1])}dK[1]\right ) (-d+c \cos (\mu K[2])+y(x))}{c \mu (b+a \cos (\lambda K[2])) (d+y(x))}dK[2]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}\frac {2 d-c \cos (\mu K[1])}{b+a \cos (\lambda K[1])}dK[1]\right )}{c \mu (b+a \cos (\lambda K[2])) (d+K[3])}-\frac {\exp \left (-\int _1^{K[2]}\frac {2 d-c \cos (\mu K[1])}{b+a \cos (\lambda K[1])}dK[1]\right ) (-d+c \cos (\mu K[2])+K[3])}{c \mu (b+a \cos (\lambda K[2])) (d+K[3])^2}\right )dK[2]-\frac {\exp \left (-\int _1^x\frac {2 d-c \cos (\mu K[1])}{b+a \cos (\lambda K[1])}dK[1]\right )}{c \mu (d+K[3])^2}\right )dK[3]=c_1,y(x)\right ]
\]