60.3.134 problem 1148
Internal
problem
ID
[11130]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
2,
linear
second
order
Problem
number
:
1148
Date
solved
:
Wednesday, March 05, 2025 at 01:43:03 PM
CAS
classification
:
[[_Emden, _Fowler]]
\begin{align*} x^{2} y^{\prime \prime }+a y&=0 \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 35
ode:=x^2*diff(diff(y(x),x),x)+a*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \sqrt {x}\, \left (c_{1} x^{\frac {\sqrt {-4 a +1}}{2}}+c_{2} x^{-\frac {\sqrt {-4 a +1}}{2}}\right )
\]
✓ Mathematica. Time used: 0.034 (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]
\[
y(x)\to x^{\frac {1}{2}-\frac {1}{2} \sqrt {1-4 a}} \left (c_2 x^{\sqrt {1-4 a}}+c_1\right )
\]
✓ Sympy. Time used: 0.454 (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 )
\]