82.50.1 problem Ex. 3

Internal problem ID [19007]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IX. Equations of the second order. problems at page 113
Problem number : Ex. 3
Date solved : Tuesday, January 28, 2025 at 12:45:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} y^{\prime \prime }+\left (-6 x^{2}+2\right ) y^{\prime }-4 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.097 (sec). Leaf size: 107

dsolve(3*x^2*diff(y(x),x$2)+(2-6*x^2)*diff(y(x),x)-4*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = \sqrt {x}\, \left (\operatorname {HeunD}\left (\frac {8 \sqrt {3}}{3}, -1-\frac {8 \sqrt {3}}{3}, \frac {16 \sqrt {3}}{3}, -\frac {8 \sqrt {3}}{3}+1, \frac {\sqrt {3}\, x -1}{\sqrt {3}\, x +1}\right ) {\mathrm e}^{\frac {2}{3 x}} c_{1} +\operatorname {HeunD}\left (-\frac {8 \sqrt {3}}{3}, -1-\frac {8 \sqrt {3}}{3}, \frac {16 \sqrt {3}}{3}, -\frac {8 \sqrt {3}}{3}+1, \frac {\sqrt {3}\, x -1}{\sqrt {3}\, x +1}\right ) {\mathrm e}^{2 x} c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.277 (sec). Leaf size: 39

DSolve[3*x^2*D[y[x],{x,2}]+(2-6*x^2)*D[y[x],x]-4*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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