23.3.301 problem 303

Internal problem ID [6015]
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 : 303
Date solved : Tuesday, September 30, 2025 at 02:19:54 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 37

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

\[ y = \frac {2 \ln \left (1+x \right ) \left (1+x \right )^{2}-3 x^{2}+\left (4 c_1 -6\right ) x +4 c_2 -1}{4 x^{2}} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 41

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

\begin{align*} y(x)&\to \frac {-3 x^2+2 (x+1)^2 \log (x+1)+(-2+4 c_2) x+4 c_1}{4 x^2} \end{align*}

✓ Sympy. Time used: 0.253 (sec). Leaf size: 42

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

\[ y{\left (x \right )} = \frac {C_{1}}{x^{2}} + \frac {C_{2}}{x} + \frac {\log {\left (x + 1 \right )}}{2} - \frac {3}{4} + \frac {\log {\left (x + 1 \right )}}{x} - \frac {1}{2 x} + \frac {\log {\left (x + 1 \right )}}{2 x^{2}} \]