Internal problem ID [10281]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 43
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)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{-\frac {x^{2}}{4}} \WhittakerM \left (-\frac {1}{8}, 0, \frac {x^{2}}{2}\right )}{\sqrt {x}}+\frac {c_{2} {\mathrm e}^{-\frac {x^{2}}{4}} \WhittakerW \left (-\frac {1}{8}, 0, \frac {x^{2}}{2}\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.06 (sec). Leaf size: 60
DSolve[4*x^2*y''[x]+4*x^3*y'[x]+(x^2+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 G_{1,2}^{2,0}\left (\frac {x^2}{16}| {c} \frac {7}{8} \\ \frac {1}{4},\frac {1}{4} \\ \\ \right )+\frac {1}{2} \sqrt [4]{-1} c_1 \sqrt {x} \text {Hypergeometric1F1}\left (\frac {3}{8},1,-\frac {x^2}{16}\right ) \\ \end{align*}