75.14.10 problem 336

Internal problem ID [16845]
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 : 336
Date solved : Thursday, March 13, 2025 at 08:53:09 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 13

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

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

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 19

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

\[ y(x)\to c_1 (-x)+c_1 x \log (x)+c_2 \]

✓ Sympy. Time used: 0.308 (sec). Leaf size: 80

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

\[ y{\left (x \right )} = C_{1} + x^{\frac {\left (\log {\left (\left |{x}\right | \right )} + 1\right ) \log {\left (\left |{x}\right | \right )} + \arg ^{2}{\left (x \right )}}{\log {\left (\left |{x}\right | \right )}^{2} + \arg ^{2}{\left (x \right )}}} \left (C_{2} \sin {\left (\log {\left (x \right )} \left |{\frac {\arg {\left (x \right )}}{\log {\left (\left |{x}\right | \right )}^{2} + \arg ^{2}{\left (x \right )}}}\right | \right )} + C_{3} \cos {\left (\frac {\log {\left (x \right )} \arg {\left (x \right )}}{\log {\left (\left |{x}\right | \right )}^{2} + \arg ^{2}{\left (x \right )}} \right )}\right ) \]