83.36.9 problem 9

Internal problem ID [19364]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise VIII (A) at page 125
Problem number : 9
Date solved : Thursday, March 13, 2025 at 02:20:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 14

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

\[ y \left (x \right ) = \left (\left (x +c_{1} \right ) {\mathrm e}^{x}+c_{2} \right ) x \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 19

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

\[ y(x)\to x \left (e^x (x-1+c_2)+c_1\right ) \]

✗ Sympy

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

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