62.30.7 problem Ex 7

Internal problem ID [12900]
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 53. Change of dependent variable. Page 125
Problem number : Ex 7
Date solved : Monday, March 31, 2025 at 07:24:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.037 (sec). Leaf size: 35

ode:=4*x^2*diff(diff(y(x),x),x)+4*x^3*diff(y(x),x)+(x^2+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-\frac {x^{2}}{4}} \left (\operatorname {WhittakerM}\left (-\frac {1}{8}, 0, \frac {x^{2}}{2}\right ) c_1 +\operatorname {WhittakerW}\left (-\frac {1}{8}, 0, \frac {x^{2}}{2}\right ) c_2 \right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.177 (sec). Leaf size: 60

ode=4*x^2*D[y[x],{x,2}]+4*x^3*D[y[x],x]+(x^2+1)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_2 G_{1,2}^{2,0}\left (\frac {x^2}{16}| \begin {array}{c} \frac {7}{8} \\ \frac {1}{4},\frac {1}{4} \\ \end {array} \right )+\frac {1}{2} \sqrt [4]{-1} c_1 \sqrt {x} \operatorname {Hypergeometric1F1}\left (\frac {3}{8},1,-\frac {x^2}{16}\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**3*Derivative(y(x), x) + 4*x**2*Derivative(y(x), (x, 2)) + (x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False