59.1.141 problem 143

Internal problem ID [9313]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 143
Date solved : Monday, January 27, 2025 at 06:01:21 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+2 x^{3} y^{\prime }+\left (3 x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 25

dsolve(4*x^2*diff(y(x),x$2)+2*x^3*diff(y(x),x)+(1+3*x^2)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.193 (sec). Leaf size: 44

DSolve[4*x^2*D[y[x],{x,2}]+2*x^3*D[y[x],x]+(1+3*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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