6.8.17 problem 20

Internal problem ID [1753]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.1 Homogeneous linear equations. Page 203
Problem number : 20
Date solved : Friday, October 03, 2025 at 01:20:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (3 x -1\right ) y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }+\left (6 x -8\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 161

ode:=(3*x-1)*diff(diff(y(x),x),x)-(3*x+2)*diff(y(x),x)+(6*x-8)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {6615 \left (c_1 \left (\left (-\frac {23 x}{35}+\frac {296}{735}\right ) \sqrt {7}+i x -\frac {248 i}{105}\right ) \operatorname {KummerM}\left (\frac {1}{2}-\frac {5 i \sqrt {7}}{14}, 3, \frac {i \sqrt {7}\, \left (3 x -1\right )}{3}\right )-c_2 \left (\left (-\frac {x}{5}-\frac {8}{105}\right ) \sqrt {7}+i x -\frac {152 i}{105}\right ) \operatorname {KummerU}\left (\frac {1}{2}-\frac {5 i \sqrt {7}}{14}, 3, \frac {i \sqrt {7}\, \left (3 x -1\right )}{3}\right )+2 \left (i+\frac {3 \sqrt {7}}{49}\right ) c_1 \operatorname {KummerM}\left (-\frac {1}{2}-\frac {5 i \sqrt {7}}{14}, 3, \frac {i \sqrt {7}\, \left (3 x -1\right )}{3}\right )+\frac {c_2 \operatorname {KummerU}\left (-\frac {1}{2}-\frac {5 i \sqrt {7}}{14}, 3, \frac {i \sqrt {7}\, \left (3 x -1\right )}{3}\right ) \left (i+\frac {5 \sqrt {7}}{7}\right )}{5}\right ) {\mathrm e}^{-\frac {x \left (i \sqrt {7}-1\right )}{2}} \left (x -\frac {1}{3}\right )^{2}}{-225 \sqrt {7}+21 i} \]

✓ Mathematica. Time used: 0.067 (sec). Leaf size: 109

ode=(3*x-1)*D[y[x],{x,2}]-(3*x+2)*D[y[x],x]+(6*x-8)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 4 e^{\frac {1}{6} \left (1-i \sqrt {7}\right ) (3 x-1)} (1-3 x)^2 \left (c_1 \operatorname {HypergeometricU}\left (\frac {3}{2}-\frac {5 i}{2 \sqrt {7}},3,\frac {1}{3} i \sqrt {7} (3 x-1)\right )+c_2 L_{-\frac {3}{2}+\frac {5 i}{2 \sqrt {7}}}^2\left (\frac {1}{3} i \sqrt {7} (3 x-1)\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((3*x - 1)*Derivative(y(x), (x, 2)) - (3*x + 2)*Derivative(y(x), x) + (6*x - 8)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False