59.1.257 problem 260

Internal problem ID [9429]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 260
Date solved : Monday, January 27, 2025 at 06:02:43 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{4} y^{\prime \prime }+x y^{\prime }+y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 50

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

\[ y = \frac {c_{1} \sqrt {2}\, \sqrt {\pi }\, \left (x -1\right ) \left (x +1\right ) \operatorname {erfi}\left (\frac {\sqrt {2}}{2 x}\right )+c_{2} x^{2}+2 \,{\mathrm e}^{\frac {1}{2 x^{2}}} c_{1} x -c_{2}}{x} \]

✓ Solution by Mathematica

Time used: 0.430 (sec). Leaf size: 57

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

\[ y(x)\to \frac {e^{3/2} \left (x^2-1\right ) \left (c_2 \int _1^x\frac {e^{\frac {1}{2 K[1]^2}-3} K[1]^2}{\left (K[1]^2-1\right )^2}dK[1]+c_1\right )}{x} \]