59.1.3 problem 3

Internal problem ID [9175]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 3
Date solved : Sunday, March 30, 2025 at 02:24:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+3\right ) y^{\prime \prime }-7 x y^{\prime }+16 y&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 65

ode:=(x^2+3)*diff(diff(y(x),x),x)-7*x*diff(y(x),x)+16*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = 4 c_2 \left (x^{4}-9 x^{2}+\frac {27}{8}\right ) \ln \left (\sqrt {x^{2}+3}-x \right )+\frac {5 \left (10 x^{3}-33 x \right ) c_2 \sqrt {x^{2}+3}}{6}+\left (c_1 +\frac {25 c_2}{3}\right ) \left (x^{4}-9 x^{2}+\frac {27}{8}\right ) \]

✓ Mathematica. Time used: 0.613 (sec). Leaf size: 69

ode=(x^2+3)*D[y[x],{x,2}]-7*x*D[y[x],x]+16*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{24} c_2 \left (3 \left (8 x^4-72 x^2+27\right ) \text {arcsinh}\left (\frac {x}{\sqrt {3}}\right )+5 x \sqrt {x^2+3} \left (33-10 x^2\right )\right )+c_1 \left (x^4-9 x^2+\frac {27}{8}\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-7*x*Derivative(y(x), x) + (x**2 + 3)*Derivative(y(x), (x, 2)) + 16*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False