60.3.87 problem 1101

Internal problem ID [11083]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1101
Date solved : Sunday, March 30, 2025 at 07:42:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+2 y^{\prime }+y a x&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 29

ode:=x*diff(diff(y(x),x),x)+2*diff(y(x),x)+y(x)*a*x = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_1 \sinh \left (\sqrt {-a}\, x \right )+c_2 \cosh \left (\sqrt {-a}\, x \right )}{x} \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 52

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

\[ y(x)\to \frac {2 c_1 e^{-i \sqrt {a} x}-\frac {i c_2 e^{i \sqrt {a} x}}{\sqrt {a}}}{2 x} \]

✓ Sympy. Time used: 0.208 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = \frac {C_{1} J_{\frac {1}{2}}\left (\sqrt {a} x\right ) + C_{2} Y_{\frac {1}{2}}\left (\sqrt {a} x\right )}{\sqrt {x}} \]