23.3.248 problem 250
Internal
problem
ID
[5962]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
3.
THE
DIFFERENTIAL
EQUATION
IS
LINEAR
AND
OF
SECOND
ORDER,
page
311
Problem
number
:
250
Date
solved
:
Tuesday, September 30, 2025 at 02:07:00 PM
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:=a*y(x)+x^2*diff(diff(y(x),x),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: 0.02 (sec). Leaf size: 42
ode=a*y[x] + x^2*D[y[x],{x,2}] == 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: 0.233 (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 )
\]