62.38.5 problem Ex 5

Internal problem ID [12945]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 62. Summary. Page 144
Problem number : Ex 5
Date solved : Monday, March 31, 2025 at 07:27:44 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 41

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

\[ y = \frac {12 \ln \left (x \right ) c_1 \,x^{3}+\left (-3 x^{4}+18 x^{2}-6 x +1\right ) c_1 +c_2 \,x^{3}}{\left (x -1\right )^{5}} \]

✓ Mathematica. Time used: 0.263 (sec). Leaf size: 93

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

\[ y(x)\to \exp \left (\int _1^x\frac {2}{K[1]-K[1]^2}dK[1]-\frac {1}{2} \int _1^x\left (\frac {6}{K[2]-1}-\frac {2}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {2}{K[1]-K[1]^2}dK[1]\right )dK[3]+c_1\right ) \]

✗ Sympy

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

False