56.32.3 problem Ex 3

Internal problem ID [14261]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VIII, Linear differential equations of the second order. Article 55. Summary. Page 129
Problem number : Ex 3
Date solved : Thursday, October 02, 2025 at 09:27:22 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 17

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

\[ y = \frac {c_1 \sinh \left (x \right )+c_2 \cosh \left (x \right )}{x^{2}} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 28

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

\begin{align*} y(x)&\to \frac {2 c_1 e^{-x}+c_2 e^x}{2 x^2} \end{align*}

✓ Sympy. Time used: 0.140 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = \frac {C_{1} J_{\frac {1}{2}}\left (i x\right ) + C_{2} Y_{\frac {1}{2}}\left (i x\right )}{x^{\frac {3}{2}}} \]