67.2.58 problem Problem 20(e)

Internal problem ID [13944]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 4, Second and Higher Order Linear Differential Equations. Problems page 221
Problem number : Problem 20(e)
Date solved : Monday, March 31, 2025 at 08:19:47 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 43

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

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

✓ Mathematica. Time used: 0.384 (sec). Leaf size: 285

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*x + (3*x + 1)*Derivative(y(x), x)/x + (x**2 - x)*Derivative(y(x), (x, 2))/x + y(x)/x,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**2 + x*Derivative(y(x), (x, 2)) - y(x))/(3*x + 1) cannot be solved by the factorable group method