61.31.2 problem 183
Internal
problem
ID
[12683]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
2,
Second-Order
Differential
Equations.
section
2.1.2-6
Problem
number
:
183
Date
solved
:
Tuesday, January 28, 2025 at 08:10:55 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{3} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c x y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.463 (sec). Leaf size: 120
dsolve(x^3*diff(y(x),x$2)+(a*x^2+b)*diff(y(x),x)+c*x*y(x)=0,y(x), singsol=all)
\[
y = x^{-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}-\frac {a}{2}+\frac {1}{2}} \left (\operatorname {KummerU}\left (-\frac {1}{4}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}+\frac {a}{4}, 1+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}, \frac {b}{2 x^{2}}\right ) c_{2} +\operatorname {KummerM}\left (-\frac {1}{4}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}+\frac {a}{4}, 1+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}, \frac {b}{2 x^{2}}\right ) c_{1} \right )
\]
✓ Solution by Mathematica
Time used: 0.389 (sec). Leaf size: 308
DSolve[x^3*D[y[x],{x,2}]+(a*x^2+b)*D[y[x],x]+c*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to -(-1)^{\frac {1}{4} \left (-\sqrt {a^2-2 a-4 c+1}+a+3\right )} 2^{\frac {1}{4} \left (-\sqrt {a^2-2 a-4 c+1}-a+1\right )} b^{\frac {1}{4} \left (-\sqrt {a^2-2 a-4 c+1}+a-1\right )} \left (\frac {1}{x}\right )^{\frac {1}{2} \left (-\sqrt {a^2-2 a-4 c+1}+a-1\right )} \left (c_2 i^{\sqrt {a^2-2 a-4 c+1}} b^{\frac {1}{2} \sqrt {a^2-2 a-4 c+1}} \left (\frac {1}{x}\right )^{\sqrt {a^2-2 a-4 c+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (a+\sqrt {a^2-2 a-4 c+1}-1\right ),\frac {1}{2} \left (\sqrt {a^2-2 a-4 c+1}+2\right ),\frac {b}{2 x^2}\right )+c_1 2^{\frac {1}{2} \sqrt {a^2-2 a-4 c+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (a-\sqrt {a^2-2 a-4 c+1}-1\right ),1-\frac {1}{2} \sqrt {a^2-2 a-4 c+1},\frac {b}{2 x^2}\right )\right )
\]