61.2.11 problem 11
Internal
problem
ID
[12017]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
1.2.2.
Equations
Containing
Power
Functions
Problem
number
:
11
Date
solved
:
Monday, January 27, 2025 at 11:50:04 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=\left (a \,x^{2 n}+b \,x^{n -1}\right ) y^{2}+c \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 1199
dsolve(diff(y(x),x)=(a*x^(2*n)+b*x^(n-1))*y(x)^2+c,y(x), singsol=all)
\[
\text {Expression too large to display}
\]
✓ Solution by Mathematica
Time used: 1.303 (sec). Leaf size: 1384
DSolve[D[y[x],x]==(a*x^(2*n)+b*x^(n-1))*y[x]^2+c,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt {c} (n+1)^2 x^{-n} \left (L_{-\frac {\sqrt {c} b}{2 \sqrt {a} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )}{\sqrt {a} c_1 (n+1) \sqrt {-(n+1)^2} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 \left (\sqrt {a} \sqrt {-(n+1)^2} n+b \sqrt {c} (n+1)\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {3 n+2}{n+1}\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+\sqrt {a} (n+1) \sqrt {-(n+1)^2} \left (L_{-\frac {\sqrt {c} b}{2 \sqrt {a} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+2 L_{-\frac {\sqrt {c} b}{2 \sqrt {a} \sqrt {-(n+1)^2}}-\frac {3 n+2}{2 n+2}}^{\frac {n}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )} \\
y(x)\to \frac {\sqrt {c} (n+1)^2 x^{-n} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}{\sqrt {a} (n+1) \sqrt {-(n+1)^2} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+\left (\sqrt {a} \sqrt {-(n+1)^2} n+b \sqrt {c} (n+1)\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}+2\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )} \\
y(x)\to \frac {\sqrt {c} (n+1)^2 x^{-n} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}{\sqrt {a} (n+1) \sqrt {-(n+1)^2} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+\left (\sqrt {a} \sqrt {-(n+1)^2} n+b \sqrt {c} (n+1)\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}+2\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )} \\
y(x)\to \frac {\sqrt {c} (n+1) x^{-n} L_{-\frac {\sqrt {c} b}{2 \sqrt {a} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}{\sqrt {a} \sqrt {-(n+1)^2} \left (2 L_{-\frac {\sqrt {c} b}{2 \sqrt {a} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}-1}^{\frac {n}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+L_{-\frac {\sqrt {c} b}{2 \sqrt {a} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )} \\
\end{align*}