69.20.3 problem 638

Internal problem ID [18424]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 638
Date solved : Thursday, October 02, 2025 at 03:11:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 15

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

\[ y = c_2 \,x^{3}+c_1 x +c_1 +1 \]

✓ Mathematica. Time used: 0.127 (sec). Leaf size: 164

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

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

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (x**2*Derivative(y(x), (x, 2))/3 + x*Derivative(y(x), (x, 2))/2 + y(x) - 1)/(x + 1) cannot be solved by the factorable group method