61.15.7 problem 16
Internal
problem
ID
[12249]
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
:
16
Date
solved
:
Tuesday, January 28, 2025 at 07:49:26 PM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
\begin{align*} y^{\prime }&=\lambda \arccos \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1} \end{align*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 165
dsolve(diff(y(x),x)=lambda*arccos(x)^n*(y(x)-a*x^m-b)^2+a*m*x^(m-1),y(x), singsol=all)
\[
y = \frac {\left (a \,x^{m}+b \right ) \lambda \left (\left (n +2\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )-\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+\arccos \left (x \right )^{n +\frac {3}{2}}\right ) \sqrt {-x^{2}+1}-\left (n +2\right ) \left (\lambda \arccos \left (x \right ) \left (a \,x^{m +1}+b x \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )-\sqrt {\arccos \left (x \right )}\, \left (x^{m} c_{1} a +c_{1} b +1\right )\right )}{\lambda \left (\left (n +2\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )-\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+\arccos \left (x \right )^{n +\frac {3}{2}}\right ) \sqrt {-x^{2}+1}+\left (n +2\right ) \left (-\arccos \left (x \right ) x \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) \lambda +c_{1} \sqrt {\arccos \left (x \right )}\right )}
\]
✓ Solution by Mathematica
Time used: 0.889 (sec). Leaf size: 44
DSolve[D[y[x],x]==\[Lambda]*ArcCos[x]^n*(y[x]-a*x^m-b)^2+a*m*x^(m-1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {1}{-\int _1^x\lambda \arccos (K[2])^ndK[2]+c_1}+a x^m+b \\
y(x)\to a x^m+b \\
\end{align*}