23.3.294 problem 296

Internal problem ID [6008]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 296
Date solved : Tuesday, September 30, 2025 at 02:19:46 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 32

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

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

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 33

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

\begin{align*} y(x)&\to a-\frac {x}{2}+\frac {c_1}{x}+\left (\frac {x}{4}+\frac {c_2}{x}\right ) \log (x) \end{align*}

✓ Sympy. Time used: 0.184 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = \frac {C_{1}}{x} + \frac {C_{2} \log {\left (x \right )}}{x} + a + \frac {x \log {\left (x \right )}}{4} - \frac {x}{2} \]