60.3.92 problem 1106

Internal problem ID [11088]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1106
Date solved : Sunday, March 30, 2025 at 07:42:12 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }+a y^{\prime }+b \,x^{\operatorname {a1}} y&=0 \end{align*}

✓ Maple. Time used: 0.064 (sec). Leaf size: 71

ode:=x*diff(diff(y(x),x),x)+a*diff(y(x),x)+b*x^a1*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\operatorname {BesselY}\left (\frac {a -1}{\operatorname {a1} +1}, \frac {2 \sqrt {b}\, x^{\frac {\operatorname {a1}}{2}+\frac {1}{2}}}{\operatorname {a1} +1}\right ) c_2 +\operatorname {BesselJ}\left (\frac {a -1}{\operatorname {a1} +1}, \frac {2 \sqrt {b}\, x^{\frac {\operatorname {a1}}{2}+\frac {1}{2}}}{\operatorname {a1} +1}\right ) c_1 \right ) x^{-\frac {a}{2}+\frac {1}{2}} \]

✓ Mathematica. Time used: 0.131 (sec). Leaf size: 165

ode=b*x^a1*y[x] + a*D[y[x],x] + x*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \left (\frac {1}{\text {a1}}+1\right )^{\frac {a-1}{\text {a1}+1}} \text {a1}^{\frac {a-1}{\text {a1}+1}} b^{\frac {1-a}{2 \text {a1}+2}} \left (x^{\text {a1}}\right )^{-\frac {a-1}{2 \text {a1}}} \left (c_2 \operatorname {Gamma}\left (\frac {-a+\text {a1}+2}{\text {a1}+1}\right ) \operatorname {BesselJ}\left (\frac {1-a}{\text {a1}+1},\frac {2 \sqrt {b} \left (x^{\text {a1}}\right )^{\frac {\text {a1}+1}{2 \text {a1}}}}{\text {a1}+1}\right )+c_1 \operatorname {Gamma}\left (\frac {a+\text {a1}}{\text {a1}+1}\right ) \operatorname {BesselJ}\left (\frac {a-1}{\text {a1}+1},\frac {2 \sqrt {b} \left (x^{\text {a1}}\right )^{\frac {\text {a1}+1}{2 \text {a1}}}}{\text {a1}+1}\right )\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
a1 = symbols("a1") 
b = symbols("b") 
y = Function("y") 
ode = Eq(a*Derivative(y(x), x) + b*x**a1*y(x) + x*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : invalid input: 1 - a