75.20.4 problem 639

Internal problem ID [17138]
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 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 639
Date solved : Tuesday, January 28, 2025 at 09:54:05 AM
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&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x \end{align*}

✓ Solution by Maple

Time used: 0.409 (sec). Leaf size: 12

dsolve([x^2*(ln(x)-1)*diff(y(x),x$2)-x*diff(y(x),x)+y(x)=0,x],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.050 (sec). Leaf size: 16

DSolve[x^2*(Log[x]-1)*D[y[x],{x,2}]-x*D[y[x],x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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