23.3.270 problem 272

Internal problem ID [5984]
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 : 272
Date solved : Tuesday, September 30, 2025 at 02:07:20 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} -a^{2} y+x y^{\prime }+x^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 17

ode:=-a^2*y(x)+x*diff(y(x),x)+x^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,x^{a}+c_2 \,x^{-a} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 25

ode=-(a^2*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 \cosh (a \log (x))+i c_2 \sinh (a \log (x)) \end{align*}

✓ Sympy. Time used: 0.124 (sec). Leaf size: 56

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a**2*y(x) + x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x^{\operatorname {re}{\left (a\right )}} \left (C_{3} \sin {\left (\log {\left (x \right )} \left |{\operatorname {im}{\left (a\right )}}\right | \right )} + C_{4} \cos {\left (\log {\left (x \right )} \operatorname {im}{\left (a\right )} \right )}\right ) + x^{- \operatorname {re}{\left (a\right )}} \left (C_{1} \sin {\left (\log {\left (x \right )} \left |{\operatorname {im}{\left (a\right )}}\right | \right )} + C_{2} \cos {\left (\log {\left (x \right )} \operatorname {im}{\left (a\right )} \right )}\right ) \]