59.1.225 problem 228

Internal problem ID [9397]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 228
Date solved : Sunday, March 30, 2025 at 02:33:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}+1\right ) y^{\prime \prime }+7 x y^{\prime }+2 y&=0 \end{align*}

✓ Maple. Time used: 0.071 (sec). Leaf size: 37

ode:=(2*x^2+1)*diff(diff(y(x),x),x)+7*x*diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_1 \operatorname {LegendreP}\left (\frac {1}{4}, \frac {3}{4}, i \sqrt {2}\, x \right )+c_2 \operatorname {LegendreQ}\left (\frac {1}{4}, \frac {3}{4}, i \sqrt {2}\, x \right )}{\left (2 x^{2}+1\right )^{{3}/{8}}} \]

✓ Mathematica. Time used: 0.062 (sec). Leaf size: 66

ode=(1+2*x^2)*D[y[x],{x,2}]+7*x*D[y[x],x]+2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {c_2 Q_{\frac {1}{4}}^{\frac {3}{4}}\left (i \sqrt {2} x\right )}{\left (2 x^2+1\right )^{3/8}}+\frac {2 i \sqrt [4]{2} c_1 x}{\left (2 x^2+1\right )^{3/4} \operatorname {Gamma}\left (\frac {1}{4}\right )} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(7*x*Derivative(y(x), x) + (2*x**2 + 1)*Derivative(y(x), (x, 2)) + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False