61.15.4 problem 13
Internal
problem
ID
[12246]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.7-2.
Equations
containing
arccosine.
Problem
number
:
13
Date
solved
:
Tuesday, January 28, 2025 at 07:49:12 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=\lambda \arccos \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arccos \left (x \right )^{n} \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 375
dsolve(diff(y(x),x)=lambda*arccos(x)^n*y(x)^2+a*y(x)+a*b-b^2*lambda*arccos(x)^n,y(x), singsol=all)
\[
\frac {-\lambda \left (\int _{}^{x}-\frac {8 \,{\mathrm e}^{\frac {2 \left (\left (n +2\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_a} \right )\right )-\arccos \left (\textit {\_a} \right ) \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\textit {\_a} \right )\right )+\arccos \left (\textit {\_a} \right )^{n +\frac {3}{2}}\right ) \lambda b \sqrt {-\textit {\_a}^{2}+1}+\textit {\_a} \left (n +2\right ) \left (-2 \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_a} \right )\right ) b \lambda \arccos \left (\textit {\_a} \right )+a \sqrt {\arccos \left (\textit {\_a} \right )}\right )}{\sqrt {\arccos \left (\textit {\_a} \right )}\, \left (n +2\right )}} \left (\left (n \left (\arcsin \left (\textit {\_a} \right )-\frac {\pi }{2}\right )^{2} b \left (n +2\right ) \operatorname {LommelS1}\left (n -\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_a} \right )\right )-b \arccos \left (\textit {\_a} \right ) \left (n +2\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_a} \right )\right )+\left (\arcsin \left (\textit {\_a} \right )-\frac {\pi }{2}\right )^{2} b \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\textit {\_a} \right )\right )+\frac {\left (\left (n +2\right ) y+b n \right ) \arccos \left (\textit {\_a} \right )^{n +\frac {5}{2}}}{2}\right ) \sqrt {-\textit {\_a}^{2}+1}+b \textit {\_a} \left (n \left (n +2\right ) \left (-\frac {\pi ^{3}}{8}+\frac {3 \arcsin \left (\textit {\_a} \right ) \arccos \left (\textit {\_a} \right ) \pi }{2}+\arcsin \left (\textit {\_a} \right )^{3}\right ) \operatorname {LommelS1}\left (n -\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_a} \right )\right )+\left (\arcsin \left (\textit {\_a} \right )-\frac {\pi }{2}\right )^{2} \left (n +2\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (\textit {\_a} \right )\right )+\left (-\frac {\pi ^{3}}{8}+\frac {3 \arcsin \left (\textit {\_a} \right ) \arccos \left (\textit {\_a} \right ) \pi }{2}+\arcsin \left (\textit {\_a} \right )^{3}\right ) \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (\textit {\_a} \right )\right )+\arccos \left (\textit {\_a} \right )^{\frac {7}{2}+n}\right )\right )}{\arccos \left (\textit {\_a} \right )^{{5}/{2}} \sqrt {-\textit {\_a}^{2}+1}}d \textit {\_a} \right )+4 \left ({\mathrm e}^{\frac {2 \left (\left (n +2\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )-\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+\arccos \left (x \right )^{n +\frac {3}{2}}\right ) \lambda b \sqrt {-x^{2}+1}+x \left (n +2\right ) \left (-2 \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) b \lambda \arccos \left (x \right )+a \sqrt {\arccos \left (x \right )}\right )}{\sqrt {\arccos \left (x \right )}\, \left (n +2\right )}}+c_{1} \left (y+b \right )\right ) \left (n +2\right )}{4 \left (n +2\right ) \left (y+b \right )} = 0
\]
✓ Solution by Mathematica
Time used: 1.639 (sec). Leaf size: 250
DSolve[D[y[x],x]==\[Lambda]*ArcCos[x]^n*y[x]^2+a*y[x]+a*b-b^2*\[Lambda]*ArcCos[x]^n,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^x-\frac {i \exp \left (-\int _1^{K[2]}\left (2 b \lambda \arccos (K[1])^n-a\right )dK[1]\right ) \left (-b \lambda \arccos (K[2])^n+\lambda y(x) \arccos (K[2])^n+a\right )}{a n \lambda (b+y(x))}dK[2]+\int _1^{y(x)}\left (\frac {i \exp \left (-\int _1^x\left (2 b \lambda \arccos (K[1])^n-a\right )dK[1]\right )}{a n \lambda (b+K[3])^2}-\int _1^x\left (\frac {i \exp \left (-\int _1^{K[2]}\left (2 b \lambda \arccos (K[1])^n-a\right )dK[1]\right ) \left (-b \lambda \arccos (K[2])^n+\lambda K[3] \arccos (K[2])^n+a\right )}{a n \lambda (b+K[3])^2}-\frac {i \exp \left (-\int _1^{K[2]}\left (2 b \lambda \arccos (K[1])^n-a\right )dK[1]\right ) \arccos (K[2])^n}{a n (b+K[3])}\right )dK[2]\right )dK[3]=c_1,y(x)\right ]
\]