47.4.6 problem 54

Internal problem ID [7483]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 2. Linear homogeneous equations. Section 2.2 problems. page 95
Problem number : 54
Date solved : Monday, January 27, 2025 at 03:01:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-x^{2}+1\right ) y^{\prime }-2 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 47

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

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

✓ Solution by Mathematica

Time used: 0.062 (sec). Leaf size: 56

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

\[ y(x)\to \frac {-4 c_1 x^2-c_2 \left (x^2-1\right ) \log (1-x)+c_2 \left (x^2-1\right ) \log (x+1)+2 c_2 x+4 c_1}{4 x^2} \]