60.3.359 problem 1376

Internal problem ID [11355]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1376
Date solved : Sunday, March 30, 2025 at 08:18:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 81

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

\[ y = \left (c_2 2^{\frac {2 i \sqrt {b}}{\sqrt {a}}} {\left (\frac {a +\sqrt {a}\, \sqrt {x^{2}+a}}{x}\right )}^{\frac {2 i \sqrt {b}}{\sqrt {a}}}+c_1 \right ) 2^{-\frac {i \sqrt {b}}{\sqrt {a}}} {\left (\frac {a +\sqrt {a}\, \sqrt {x^{2}+a}}{x}\right )}^{-\frac {i \sqrt {b}}{\sqrt {a}}} \]

✓ Mathematica. Time used: 0.144 (sec). Leaf size: 69

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

\[ y(x)\to c_1 \cos \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {a+x^2}}{\sqrt {a}}\right )}{\sqrt {a}}\right )-c_2 \sin \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {a+x^2}}{\sqrt {a}}\right )}{\sqrt {a}}\right ) \]

✗ Sympy

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

ValueError : Expected Expr or iterable but got None