64.12.21 problem 21

Internal problem ID [13454]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.4. Variation of parameters. Exercises page 162
Problem number : 21
Date solved : Monday, March 31, 2025 at 07:58:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 24

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

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

✓ Mathematica. Time used: 0.232 (sec). Leaf size: 266

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

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

✗ Sympy

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