83.49.20 problem Ex 20 page 135

Internal problem ID [19633]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter VIII. Linear equations of second order
Problem number : Ex 20 page 135
Date solved : Tuesday, January 28, 2025 at 02:05:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 107

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

\[ y = \frac {c_1 \,{\mathrm e}^{\frac {2}{3 x}} \operatorname {HeunD}\left (\frac {8 \sqrt {3}}{3}, -\frac {8 \sqrt {3}}{3}-1, -\frac {16 \sqrt {3}}{3}, 1-\frac {8 \sqrt {3}}{3}, \frac {\sqrt {3}\, x -1}{\sqrt {3}\, x +1}\right )+c_2 \,{\mathrm e}^{2 x} \operatorname {HeunD}\left (-\frac {8 \sqrt {3}}{3}, -\frac {8 \sqrt {3}}{3}-1, -\frac {16 \sqrt {3}}{3}, 1-\frac {8 \sqrt {3}}{3}, \frac {\sqrt {3}\, x -1}{\sqrt {3}\, x +1}\right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.316 (sec). Leaf size: 48

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

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