61.2.38 problem 38
Internal
problem
ID
[12044]
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
:
38
Date
solved
:
Monday, January 27, 2025 at 11:53:18 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} x y^{\prime }+a_{3} x y^{2}+a_{2} y+a_{1} x +a_{0}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.043 (sec). Leaf size: 403
dsolve(x*diff(y(x),x)+a__3*x*y(x)^2+a__2*y(x)+a__1*x+a__0=0,y(x), singsol=all)
\[
y = -\frac {4 a_{1} \left (a_{1}^{3} a_{3} \left (a_{0} a_{3} -a_{2} \sqrt {-a_{1} a_{3}}\right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )-\frac {c_{1} \left (a_{0}^{2} a_{3} +a_{1} a_{2}^{2}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )}{4}+a_{1}^{3} a_{3} \left (a_{2} \sqrt {-a_{1} a_{3}}+a_{0} a_{3} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )+\frac {\operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right ) c_{1} \left (\sqrt {-a_{1} a_{3}}\, a_{0} -a_{1} a_{2} \right )}{2}\right )}{4 a_{1}^{3} a_{3}^{2} \left (\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )-c_{1} \sqrt {-a_{1} a_{3}}\, \left (a_{0}^{2} a_{3} +a_{1} a_{2}^{2}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )+2 a_{1} \left (-2 a_{1}^{2} a_{3}^{2} \left (\sqrt {-a_{1} a_{3}}\, a_{0} -a_{1} a_{2} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )+\operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right ) c_{1} \left (a_{2} \sqrt {-a_{1} a_{3}}+a_{0} a_{3} \right )\right )}
\]
✓ Solution by Mathematica
Time used: 0.414 (sec). Leaf size: 421
DSolve[x*D[y[x],x]+a3*x*y[x]^2+a2*y[x]+a1*x+a0==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {i \left (\sqrt {\text {a1}} c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+c_1 \left (\sqrt {\text {a1}} \text {a2}+i \text {a0} \sqrt {\text {a3}}\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}+2\right ),\text {a2}+1,2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+\sqrt {\text {a1}} \left (2 L_{-\frac {i \sqrt {\text {a3}} \text {a0}}{2 \sqrt {\text {a1}}}-\frac {\text {a2}}{2}-1}^{\text {a2}}\left (2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+L_{-\frac {i \sqrt {\text {a3}} \text {a0}}{2 \sqrt {\text {a1}}}-\frac {\text {a2}}{2}}^{\text {a2}-1}\left (2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )\right )\right )}{\sqrt {\text {a3}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+L_{-\frac {i \sqrt {\text {a3}} \text {a0}}{2 \sqrt {\text {a1}}}-\frac {\text {a2}}{2}}^{\text {a2}-1}\left (2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )\right )} \\
y(x)\to \frac {\frac {\left (\text {a0} \sqrt {\text {a3}}-i \sqrt {\text {a1}} \text {a2}\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}+2\right ),\text {a2}+1,2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )}{\operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )}-i \sqrt {\text {a1}}}{\sqrt {\text {a3}}} \\
\end{align*}