62.30.7 problem Ex 7

Internal problem ID [12971]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VIII, Linear differential equations of the second order. Article 53. Change of dependent variable. Page 125
Problem number : Ex 7
Date solved : Tuesday, January 28, 2025 at 08:24:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.604 (sec). Leaf size: 35

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

\[ y = \frac {{\mathrm e}^{-\frac {x^{2}}{4}} \left (\operatorname {WhittakerM}\left (-\frac {1}{8}, 0, \frac {x^{2}}{2}\right ) c_{1} +\operatorname {WhittakerW}\left (-\frac {1}{8}, 0, \frac {x^{2}}{2}\right ) c_{2} \right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.177 (sec). Leaf size: 60

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

\[ y(x)\to c_2 G_{1,2}^{2,0}\left (\frac {x^2}{16}| \begin {array}{c} \frac {7}{8} \\ \frac {1}{4},\frac {1}{4} \\ \end {array} \right )+\frac {1}{2} \sqrt [4]{-1} c_1 \sqrt {x} \operatorname {Hypergeometric1F1}\left (\frac {3}{8},1,-\frac {x^2}{16}\right ) \]