75.14.5 problem 331

Internal problem ID [16848]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 14. Differential equations admitting of depression of their order. Exercises page 107
Problem number : 331
Date solved : Monday, March 31, 2025 at 03:24:13 PM
CAS classification : [[_2nd_order, _quadrature]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 21

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

\[ y = -\frac {5 x^{3}}{18}+\frac {x^{3} \ln \left (x \right )}{3}+c_1 x +c_2 \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 28

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

\[ y(x)\to -\frac {5 x^3}{18}+\frac {1}{3} x^3 \log (x)+c_2 x+c_1 \]

✓ Sympy. Time used: 0.181 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = C_{1} + C_{2} x + \frac {x^{3} \log {\left (x \right )}}{3} - \frac {5 x^{3}}{18} \]