23.3.317 problem 319

Internal problem ID [6031]
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 : 319
Date solved : Tuesday, September 30, 2025 at 02:20:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} a \left (1+a \right ) y-2 a x y^{\prime }+x^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 13

ode:=a*(a+1)*y(x)-2*a*x*diff(y(x),x)+x^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{a} \left (c_1 x +c_2 \right ) \]

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 105

ode=a*(1 + a)*y[x] - 2*a*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 x^{\frac {-\sqrt {\frac {1}{a^2+a}} \sqrt {a}-\sqrt {\frac {1}{a^2+a}} a^{3/2}+2 \sqrt {a+1} a+\sqrt {a+1}}{2 \sqrt {a+1}}} \left (c_2 x^{\sqrt {a} \sqrt {a+1} \sqrt {\frac {1}{a^2+a}}}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.152 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-2*a*x*Derivative(y(x), x) + a*(a + 1)*y(x) + x**2*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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