54.3.16 problem 16

Internal problem ID [8572]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 16
Date solved : Monday, January 27, 2025 at 04:16:36 PM
CAS classification : [[_3rd_order, _exact, _linear, _homogeneous]]

\begin{align*} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }+5 x y^{\prime }+3 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 96

dsolve(diff(y(x),x$3)+x^2*diff(y(x),x$2)+5*x*diff(y(x),x)+3*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {\left (c_{1} {\mathrm e}^{-\frac {x^{3}}{3}} x +c_3 3^{{1}/{3}}\right ) \left (-x^{3}\right )^{{2}/{3}}+x^{2} {\mathrm e}^{-\frac {x^{3}}{3}} \left (-3 c_{2} \Gamma \left (\frac {1}{3}, -\frac {x^{3}}{3}\right ) \left (-x^{3}\right )^{{1}/{3}} \Gamma \left (\frac {2}{3}\right )+2 c_{2} \left (-x^{3}\right )^{{1}/{3}} \pi \sqrt {3}+x \Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right ) c_3 -x c_3 \Gamma \left (\frac {2}{3}\right )\right )}{\left (-x^{3}\right )^{{2}/{3}}} \]

✓ Solution by Mathematica

Time used: 0.099 (sec). Leaf size: 88

DSolve[D[y[x],{x,3}]+x^2*D[y[x],{x,2}]+5*x*D[y[x],x]+3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {e^{-\frac {x^3}{3}} \left (-2\ 3^{2/3} c_3 \sqrt [3]{-x^3} x \Gamma \left (-\frac {1}{3},-\frac {x^3}{3}\right )+3 \sqrt [3]{3} c_1 \left (-x^3\right )^{2/3} \Gamma \left (\frac {1}{3},-\frac {x^3}{3}\right )+18 c_2 x^2\right )}{18 x} \]