55.30.1 problem 110
Internal
problem
ID
[13883]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
2,
Second-Order
Differential
Equations.
section
2.1.2-4
Problem
number
:
110
Date
solved
:
Thursday, October 02, 2025 at 08:08:27 AM
CAS
classification
:
[[_Emden, _Fowler]]
\begin{align*} a y+x^{2} y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 35
ode:=x^2*diff(diff(y(x),x),x)+a*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \sqrt {x}\, \left (c_1 \,x^{\frac {\sqrt {1-4 a}}{2}}+c_2 \,x^{-\frac {\sqrt {1-4 a}}{2}}\right )
\]
✓ Mathematica. Time used: 0.022 (sec). Leaf size: 42
ode=x^2*D[y[x],{x,2}]+a*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to x^{\frac {1}{2}-\frac {1}{2} \sqrt {1-4 a}} \left (c_2 x^{\sqrt {1-4 a}}+c_1\right ) \end{align*}
✓ Sympy. Time used: 0.238 (sec). Leaf size: 267
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(a*y(x) + x**2*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = x^{- \frac {\sqrt [4]{\left (4 \operatorname {re}{\left (a\right )} - 1\right )^{2} + 16 \left (\operatorname {im}{\left (a\right )}\right )^{2}} \cos {\left (\frac {\operatorname {atan}_{2}{\left (- 4 \operatorname {im}{\left (a\right )},1 - 4 \operatorname {re}{\left (a\right )} \right )}}{2} \right )}}{2} + \frac {1}{2}} \left (C_{1} \sin {\left (\frac {\sqrt [4]{\left (4 \operatorname {re}{\left (a\right )} - 1\right )^{2} + 16 \left (\operatorname {im}{\left (a\right )}\right )^{2}} \log {\left (x \right )} \left |{\sin {\left (\frac {\operatorname {atan}_{2}{\left (- 4 \operatorname {im}{\left (a\right )},1 - 4 \operatorname {re}{\left (a\right )} \right )}}{2} \right )}}\right |}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt [4]{\left (4 \operatorname {re}{\left (a\right )} - 1\right )^{2} + 16 \left (\operatorname {im}{\left (a\right )}\right )^{2}} \log {\left (x \right )} \sin {\left (\frac {\operatorname {atan}_{2}{\left (- 4 \operatorname {im}{\left (a\right )},1 - 4 \operatorname {re}{\left (a\right )} \right )}}{2} \right )}}{2} \right )}\right ) + x^{\frac {\sqrt [4]{\left (4 \operatorname {re}{\left (a\right )} - 1\right )^{2} + 16 \left (\operatorname {im}{\left (a\right )}\right )^{2}} \cos {\left (\frac {\operatorname {atan}_{2}{\left (- 4 \operatorname {im}{\left (a\right )},1 - 4 \operatorname {re}{\left (a\right )} \right )}}{2} \right )}}{2} + \frac {1}{2}} \left (C_{3} \sin {\left (\frac {\sqrt [4]{\left (4 \operatorname {re}{\left (a\right )} - 1\right )^{2} + 16 \left (\operatorname {im}{\left (a\right )}\right )^{2}} \log {\left (x \right )} \left |{\sin {\left (\frac {\operatorname {atan}_{2}{\left (- 4 \operatorname {im}{\left (a\right )},1 - 4 \operatorname {re}{\left (a\right )} \right )}}{2} \right )}}\right |}{2} \right )} + C_{4} \cos {\left (\frac {\sqrt [4]{\left (4 \operatorname {re}{\left (a\right )} - 1\right )^{2} + 16 \left (\operatorname {im}{\left (a\right )}\right )^{2}} \log {\left (x \right )} \sin {\left (\frac {\operatorname {atan}_{2}{\left (- 4 \operatorname {im}{\left (a\right )},1 - 4 \operatorname {re}{\left (a\right )} \right )}}{2} \right )}}{2} \right )}\right )
\]