23.3.130 problem 132

Internal problem ID [5844]
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 : 132
Date solved : Tuesday, September 30, 2025 at 02:04:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.097 (sec). Leaf size: 140

ode:=a*(1+k)*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);
 

\[ y = c_1 x \,{\mathrm e}^{-\frac {x a \,x^{k}}{1+k}}+\frac {c_2 \,x^{-\frac {3 k}{2}} {\mathrm e}^{-\frac {a \,x^{k} x}{2+2 k}} \left (\left (1+k \right ) \left (a \,x^{k} x -k \right ) \operatorname {WhittakerM}\left (\frac {-2-k}{2+2 k}, \frac {2 k +1}{2+2 k}, -\frac {x a \,x^{k}}{1+k}\right )-\operatorname {WhittakerM}\left (\frac {k}{2+2 k}, \frac {2 k +1}{2+2 k}, -\frac {x a \,x^{k}}{1+k}\right ) k^{2}\right )}{x} \]

✗ Mathematica

ode=a*(1 + k)*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]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
k = symbols("k") 
y = Function("y") 
ode = Eq(a*x**k*Derivative(y(x), x) + a*x**(k - 1)*(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