60.3.261 problem 1277

Internal problem ID [11257]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1277
Date solved : Wednesday, March 05, 2025 at 02:04:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+4 x y^{\prime }-\left (a \,x^{2}+1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.089 (sec). Leaf size: 27

ode:=4*x^2*diff(diff(y(x),x),x)+4*x*diff(y(x),x)-(a*x^2+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_{1} \sinh \left (\frac {\sqrt {a}\, x}{2}\right )+c_{2} \cosh \left (\frac {\sqrt {a}\, x}{2}\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.055 (sec). Leaf size: 49

ode=(-1 - a*x^2)*y[x] + 4*x*D[y[x],x] + 4*x^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.229 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + 4*x*Derivative(y(x), x) - (a*x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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