59.1.162 problem 164

Internal problem ID [9334]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 164
Date solved : Monday, January 27, 2025 at 06:01:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-x \right )^{2} y^{\prime \prime }-x \left (-3 x^{2}+2 x +1\right ) y^{\prime }+\left (x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 17

dsolve(x^2*(1-x)^2*diff(y(x),x$2)-x*(1+2*x-3*x^2)*diff(y(x),x)+(1+x^2)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.283 (sec). Leaf size: 47

DSolve[x^2*(1-x)^2*D[y[x],{x,2}]-x*(1+2*x-3*x^2)*D[y[x],x]+(1+x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \sqrt {x} (c_2 \log (x)+c_1) \exp \left (-\frac {1}{2} \int _1^x\left (\frac {4}{K[1]-1}-\frac {1}{K[1]}\right )dK[1]\right ) \]