59.1.185 problem 187

Internal problem ID [9357]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 187
Date solved : Monday, January 27, 2025 at 06:01:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (-2 x^{2}+1\right ) y^{\prime }-4 \left (2 x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 41

dsolve(x^2*diff(y(x),x$2)+x*(1-2*x^2)*diff(y(x),x)-4*(1+2*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {-{\mathrm e}^{x^{2}} \operatorname {Ei}_{1}\left (x^{2}\right ) c_{2} x^{4}+c_{1} x^{4} {\mathrm e}^{x^{2}}+c_{2} x^{2}-c_{2}}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.114 (sec). Leaf size: 46

DSolve[x^2*D[y[x],{x,2}]+x*(1-2*x^2)*D[y[x],x]-4*(1+2*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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