54.3.146 problem 1160
Internal
problem
ID
[12441]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
2,
linear
second
order
Problem
number
:
1160
Date
solved
:
Wednesday, October 01, 2025 at 01:45:03 AM
CAS
classification
:
[[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+a y&=0 \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 23
ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+a*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = c_1 \sin \left (\sqrt {a}\, \ln \left (x \right )\right )+c_2 \cos \left (\sqrt {a}\, \ln \left (x \right )\right )
\]
✓ Mathematica. Time used: 0.013 (sec). Leaf size: 30
ode=a*y[x] + x*D[y[x],x] + x^2*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to c_1 \cos \left (\sqrt {a} \log (x)\right )+c_2 \sin \left (\sqrt {a} \log (x)\right ) \end{align*}
✓ Sympy. Time used: 0.205 (sec). Leaf size: 199
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(a*y(x) + x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = x^{\sqrt [4]{\left (\operatorname {re}{\left (a\right )}\right )^{2} + \left (\operatorname {im}{\left (a\right )}\right )^{2}} \cos {\left (\frac {\operatorname {atan}_{2}{\left (- \operatorname {im}{\left (a\right )},- \operatorname {re}{\left (a\right )} \right )}}{2} \right )}} \left (C_{3} \sin {\left (\sqrt [4]{\left (\operatorname {re}{\left (a\right )}\right )^{2} + \left (\operatorname {im}{\left (a\right )}\right )^{2}} \log {\left (x \right )} \left |{\sin {\left (\frac {\operatorname {atan}_{2}{\left (- \operatorname {im}{\left (a\right )},- \operatorname {re}{\left (a\right )} \right )}}{2} \right )}}\right | \right )} + C_{4} \cos {\left (\sqrt [4]{\left (\operatorname {re}{\left (a\right )}\right )^{2} + \left (\operatorname {im}{\left (a\right )}\right )^{2}} \log {\left (x \right )} \sin {\left (\frac {\operatorname {atan}_{2}{\left (- \operatorname {im}{\left (a\right )},- \operatorname {re}{\left (a\right )} \right )}}{2} \right )} \right )}\right ) + x^{- \sqrt [4]{\left (\operatorname {re}{\left (a\right )}\right )^{2} + \left (\operatorname {im}{\left (a\right )}\right )^{2}} \cos {\left (\frac {\operatorname {atan}_{2}{\left (- \operatorname {im}{\left (a\right )},- \operatorname {re}{\left (a\right )} \right )}}{2} \right )}} \left (C_{1} \sin {\left (\sqrt [4]{\left (\operatorname {re}{\left (a\right )}\right )^{2} + \left (\operatorname {im}{\left (a\right )}\right )^{2}} \log {\left (x \right )} \left |{\sin {\left (\frac {\operatorname {atan}_{2}{\left (- \operatorname {im}{\left (a\right )},- \operatorname {re}{\left (a\right )} \right )}}{2} \right )}}\right | \right )} + C_{2} \cos {\left (\sqrt [4]{\left (\operatorname {re}{\left (a\right )}\right )^{2} + \left (\operatorname {im}{\left (a\right )}\right )^{2}} \log {\left (x \right )} \sin {\left (\frac {\operatorname {atan}_{2}{\left (- \operatorname {im}{\left (a\right )},- \operatorname {re}{\left (a\right )} \right )}}{2} \right )} \right )}\right )
\]