23.3.133 problem 135

Internal problem ID [5847]
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 : 135
Date solved : Friday, October 03, 2025 at 01:44:04 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} b \,x^{-1+k} y+a \,x^{k} y^{\prime }+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.107 (sec). Leaf size: 96

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

\[ y = x \left (\operatorname {KummerU}\left (\frac {1+k -\frac {b}{a}}{k +1}, \frac {k +2}{k +1}, \frac {x a \,x^{k}}{k +1}\right ) c_2 +\operatorname {KummerM}\left (\frac {1+k -\frac {b}{a}}{k +1}, \frac {k +2}{k +1}, \frac {x a \,x^{k}}{k +1}\right ) c_1 \right ) {\mathrm e}^{-\frac {x a \,x^{k}}{k +1}} \]

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 120

ode=b*x^(-1 + k)*y[x] + a*x^k*D[y[x],x] + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 \left (\frac {1}{k}+1\right )^{-\frac {1}{k+1}} k^{-\frac {1}{k+1}} a^{\frac {1}{k+1}} \left (x^k\right )^{\frac {1}{k}} \operatorname {Hypergeometric1F1}\left (\frac {a+b}{k a+a},\frac {k+2}{k+1},-\frac {a \left (x^k\right )^{1+\frac {1}{k}}}{k+1}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {b}{k a+a},\frac {k}{k+1},-\frac {a \left (x^k\right )^{1+\frac {1}{k}}}{k+1}\right ) \end{align*}

✗ Sympy

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

TypeError : Add object cannot be interpreted as an integer