60.3.166 problem 1180

Internal problem ID [11162]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1180
Date solved : Sunday, March 30, 2025 at 07:44:35 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (-v^{2}+x^{2}+1\right ) y-f \left (x \right )&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 49

ode:=x^2*diff(diff(y(x),x),x)+3*x*diff(y(x),x)+(-v^2+x^2+1)*y(x)-f(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-\pi \int \operatorname {BesselY}\left (v , x\right ) f \left (x \right )d x \operatorname {BesselJ}\left (v , x\right )+\pi \int \operatorname {BesselJ}\left (v , x\right ) f \left (x \right )d x \operatorname {BesselY}\left (v , x\right )+2 \operatorname {BesselY}\left (v , x\right ) c_1 +2 \operatorname {BesselJ}\left (v , x\right ) c_2}{2 x} \]

✓ Mathematica. Time used: 0.082 (sec). Leaf size: 68

ode=-f[x] + (1 - v^2 + x^2)*y[x] + 3*x*D[y[x],x] + x^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {\operatorname {BesselJ}(v,x) \int _1^x-\frac {1}{2} \pi \operatorname {BesselY}(v,K[1]) f(K[1])dK[1]+\operatorname {BesselY}(v,x) \int _1^x\frac {1}{2} \pi \operatorname {BesselJ}(v,K[2]) f(K[2])dK[2]+c_1 \operatorname {BesselJ}(v,x)+c_2 \operatorname {BesselY}(v,x)}{x} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
v = symbols("v") 
y = Function("y") 
f = Function("f") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + 3*x*Derivative(y(x), x) + (-v**2 + x**2 + 1)*y(x) - f(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (v**2*y(x) - x**2*(y(x) + Derivative(y(x), (x, 2))) + f(x) - y(x))/(3*x) cannot be solved by the factorable group method