47.5.8 problem 8

Internal problem ID [7497]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 2. Linear homogeneous equations. Section 2.3.4 problems. page 104
Problem number : 8
Date solved : Monday, January 27, 2025 at 03:02:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 25

dsolve(x^2*(ln(x)-1)*diff(y(x),x$2)-x*diff(y(x),x)+y(x)=x*(1-ln(x))^2,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.081 (sec). Leaf size: 27

DSolve[x^2*(Log[x]-1)*D[y[x],{x,2}]-x*D[y[x],x]+y[x]==x*(1-Log[x])^2,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} x \log ^2(x)+c_1 x-(x+c_2) \log (x) \]