61.32.9 problem 219
Internal
problem
ID
[12719]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
2,
Second-Order
Differential
Equations.
section
2.1.2-7
Problem
number
:
219
Date
solved
:
Tuesday, January 28, 2025 at 08:23:49 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{2} \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) x y^{\prime }+d y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.484 (sec). Leaf size: 286
dsolve(x^2*(x^2+a)*diff(y(x),x$2)+(b*x^2+c)*x*diff(y(x),x)+d*y(x)=0,y(x), singsol=all)
\[
y = \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \left (c_{2} x^{-\frac {-a +c +\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{2 a}} \operatorname {hypergeom}\left (\left [-\frac {-3 a -c +\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{4 a}, \frac {-\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}+\left (-2 b +5\right ) a +c}{4 a}\right ], \left [1-\frac {\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{2 a}\right ], -\frac {x^{2}}{a}\right )+c_{1} x^{\frac {a -c +\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{2 a}} \operatorname {hypergeom}\left (\left [\frac {3 a +c +\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{4 a}, \frac {\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}+\left (-2 b +5\right ) a +c}{4 a}\right ], \left [1+\frac {\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{2 a}\right ], -\frac {x^{2}}{a}\right )\right )
\]
✓ Solution by Mathematica
Time used: 1.410 (sec). Leaf size: 336
DSolve[x^2*(x^2+a)*D[y[x],{x,2}]+(b*x^2+c)*x*D[y[x],x]+d*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to a^{-\frac {\sqrt {a^2-2 a (c+2 d)+c^2}+a-c}{4 a}} x^{-\frac {\sqrt {a^2-2 a (c+2 d)+c^2}-a+c}{2 a}} \left (c_2 x^{\frac {\sqrt {a^2-2 a (c+2 d)+c^2}}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {-2 b a+a+c-\sqrt {a^2-2 (c+2 d) a+c^2}}{4 a},\frac {a-c+\sqrt {a^2-2 (c+2 d) a+c^2}}{4 a},\frac {\sqrt {a^2-2 (c+2 d) a+c^2}}{2 a}+1,-\frac {x^2}{a}\right )+c_1 a^{\frac {\sqrt {a^2-2 a (c+2 d)+c^2}}{2 a}} \operatorname {Hypergeometric2F1}\left (-\frac {-a+c+\sqrt {a^2-2 (c+2 d) a+c^2}}{4 a},-\frac {-2 b a+a+c+\sqrt {a^2-2 (c+2 d) a+c^2}}{4 a},1-\frac {\sqrt {a^2-2 (c+2 d) a+c^2}}{2 a},-\frac {x^2}{a}\right )\right )
\]