61.2.12 problem 12
Internal
problem
ID
[12018]
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
:
12
Date
solved
:
Monday, January 27, 2025 at 11:50:55 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} \left (a_{2} x +b_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+a_{0} x +b_{0}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.026 (sec). Leaf size: 461
dsolve((a__2*x+b__2)*(diff(y(x),x)+lambda*y(x)^2)+a__0*x+b__0=0,y(x), singsol=all)
\[
y = -\frac {a_{0} \left (\frac {c_{1} \lambda \left (a_{0} b_{2} -a_{2} b_{0} \right ) \operatorname {KummerU}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, a_{0} b_{2} -\sqrt {-a_{2} \lambda a_{0}}\, a_{2} b_{0} -2 a_{0} a_{2}^{2}}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right )}{2}+\sqrt {-a_{2} \lambda a_{0}}\, a_{2} \left (c_{1} \operatorname {KummerU}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right )+\lambda \operatorname {KummerM}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, a_{0} b_{2} -\sqrt {-a_{2} \lambda a_{0}}\, a_{2} b_{0} -2 a_{0} a_{2}^{2}}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right )+\operatorname {KummerM}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right ) \lambda \right )\right )}{\lambda \left (\frac {c_{1} \sqrt {-a_{2} \lambda a_{0}}\, \left (a_{0} b_{2} -a_{2} b_{0} \right ) \operatorname {KummerU}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, a_{0} b_{2} -\sqrt {-a_{2} \lambda a_{0}}\, a_{2} b_{0} -2 a_{0} a_{2}^{2}}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right )}{2}+a_{2}^{2} a_{0} \left (\operatorname {KummerM}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right ) \lambda +c_{1} \operatorname {KummerU}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right )-\lambda \operatorname {KummerM}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, a_{0} b_{2} -\sqrt {-a_{2} \lambda a_{0}}\, a_{2} b_{0} -2 a_{0} a_{2}^{2}}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right )\right )\right )}
\]
✓ Solution by Mathematica
Time used: 1.166 (sec). Leaf size: 540
DSolve[(a2*x+b2)*(D[y[x],x]+\[Lambda]*y[x]^2)+a0*x+b0==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {c_1 \sqrt {\lambda } (\text {a2} \text {b0}-\text {a0} \text {b2}) \operatorname {HypergeometricU}\left (\frac {i \sqrt {\lambda } (\text {a2} \text {b0}-\text {a0} \text {b2})}{2 \sqrt {\text {a0}} \text {a2}^{3/2}}+1,1,\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )-i \sqrt {\text {a0}} \text {a2}^{3/2} \left (c_1 \operatorname {HypergeometricU}\left (\frac {i (\text {a2} \text {b0}-\text {a0} \text {b2}) \sqrt {\lambda }}{2 \sqrt {\text {a0}} \text {a2}^{3/2}},0,\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )+2 \operatorname {LaguerreL}\left (\frac {i (\text {a0} \text {b2}-\text {a2} \text {b0}) \sqrt {\lambda }}{2 \sqrt {\text {a0}} \text {a2}^{3/2}}-1,\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )+L_{\frac {i (\text {a0} \text {b2}-\text {a2} \text {b0}) \sqrt {\lambda }}{2 \sqrt {\text {a0}} \text {a2}^{3/2}}}^{-1}\left (\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )\right )}{\text {a2}^2 \sqrt {\lambda } \left (c_1 \operatorname {HypergeometricU}\left (\frac {i (\text {a2} \text {b0}-\text {a0} \text {b2}) \sqrt {\lambda }}{2 \sqrt {\text {a0}} \text {a2}^{3/2}},0,\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )+L_{\frac {i (\text {a0} \text {b2}-\text {a2} \text {b0}) \sqrt {\lambda }}{2 \sqrt {\text {a0}} \text {a2}^{3/2}}}^{-1}\left (\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )\right )} \\
y(x)\to \frac {(\text {a2} \text {b0}-\text {a0} \text {b2}) \operatorname {HypergeometricU}\left (\frac {i \sqrt {\lambda } (\text {a2} \text {b0}-\text {a0} \text {b2})}{2 \sqrt {\text {a0}} \text {a2}^{3/2}}+1,1,\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )}{\text {a2}^2 \operatorname {HypergeometricU}\left (\frac {i (\text {a2} \text {b0}-\text {a0} \text {b2}) \sqrt {\lambda }}{2 \sqrt {\text {a0}} \text {a2}^{3/2}},0,\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )}-\frac {i \sqrt {\text {a0}}}{\sqrt {\text {a2}} \sqrt {\lambda }} \\
\end{align*}