69.20.11 problem 646

Internal problem ID [18432]
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 : 646
Date solved : Thursday, October 02, 2025 at 03:11:51 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} \left (x^{4}-x^{3}\right ) y^{\prime \prime }+\left (2 x^{3}-2 x^{2}-x \right ) y^{\prime }-y&=\frac {\left (x -1\right )^{2}}{x} \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {1}{x} \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 22

ode:=(x^4-x^3)*diff(diff(y(x),x),x)+(2*x^3-2*x^2-x)*diff(y(x),x)-y(x) = (x-1)^2/x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{\frac {1}{x}} c_1 x -\ln \left (x \right )+c_2 +x}{x} \]

✓ Mathematica. Time used: 0.244 (sec). Leaf size: 281

ode=(x^4-x^3)*D[y[x],{x,2}]+(2*x^3-2*x^2-x)*D[y[x],x]-y[x]==(x-1)^2/x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \exp \left (\int _1^x\frac {1}{2 K[1]^2-2 K[1]^3}dK[1]-\frac {1}{2} \int _1^x\frac {2 K[2]+\frac {1}{1-K[2]}}{K[2]^2}dK[2]\right ) \left (\int _1^x-\frac {\exp \left (\int _1^{K[4]}\frac {1}{2 K[1]^2-2 K[1]^3}dK[1]+\frac {1}{2} \int _1^{K[4]}\frac {2 K[2]+\frac {1}{1-K[2]}}{K[2]^2}dK[2]\right ) (K[4]-1) \int _1^{K[4]}\exp \left (-2 \int _1^{K[3]}\frac {1}{2 K[1]^2-2 K[1]^3}dK[1]\right )dK[3]}{K[4]^4}dK[4]+\int _1^x\exp \left (-2 \int _1^{K[3]}\frac {1}{2 K[1]^2-2 K[1]^3}dK[1]\right )dK[3] \left (\int _1^x\frac {\exp \left (\int _1^{K[5]}\frac {1}{2 K[1]^2-2 K[1]^3}dK[1]+\frac {1}{2} \int _1^{K[5]}\frac {2 K[2]+\frac {1}{1-K[2]}}{K[2]^2}dK[2]\right ) (K[5]-1)}{K[5]^4}dK[5]+c_2\right )+c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x**4 - x**3)*Derivative(y(x), (x, 2)) + (2*x**3 - 2*x**2 - x)*Derivative(y(x), x) - y(x) - (x - 1)**2/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (x**5*Derivative(y(x), (x, 2)) - x**4*Derivative(y(x), (x, 2)) - x**2 - x*y(x) + 2*x - 1)/(x**2*(-2*x**2 + 2*x + 1)) cannot be solved by the factorable group method