59.1.316 problem 321

Internal problem ID [9488]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 321
Date solved : Monday, January 27, 2025 at 06:03:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.056 (sec). Leaf size: 45

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

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

✓ Solution by Mathematica

Time used: 0.081 (sec). Leaf size: 92

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

\[ y(x)\to \frac {2^{3/4} c_1 \cos \left (2 \sqrt {2} \arcsin \left (\frac {1}{2} \sqrt {2-i \sqrt {2} x}\right )\right )}{\sqrt {\pi } \sqrt {x^2+2}}+\frac {c_2 Q_{-\frac {1}{2}+\sqrt {2}}^{\frac {1}{2}}\left (\frac {i x}{\sqrt {2}}\right )}{\sqrt [4]{x^2+2}} \]