Internal problem ID [6904]
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.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _exact, _linear, _homogeneous]]
\[ \boxed {y^{\prime \prime \prime }+x^{2} y^{\prime \prime }+5 y^{\prime } x +3 y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 101
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)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x^{3}}{3}} x +\frac {c_{2} x^{2} \left (3 \Gamma \left (\frac {1}{3}, -\frac {x^{3}}{3}\right ) \Gamma \left (\frac {2}{3}\right )-2 \sqrt {3}\, \pi \right ) {\mathrm e}^{-\frac {x^{3}}{3}}}{\left (-x^{3}\right )^{\frac {1}{3}}}+\frac {c_{3} \left (\left (-x^{3}\right )^{\frac {2}{3}} 3^{\frac {1}{3}}-\Gamma \left (\frac {2}{3}\right ) x^{3} {\mathrm e}^{-\frac {x^{3}}{3}}+\Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right ) x^{3} {\mathrm e}^{-\frac {x^{3}}{3}}\right )}{\left (-x^{3}\right )^{\frac {2}{3}}} \]
✓ Solution by Mathematica
Time used: 0.093 (sec). Leaf size: 88
DSolve[y'''[x]+x^2*y''[x]+5*x*y'[x]+3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-\frac {x^3}{3}} \left (-2\ 3^{2/3} c_3 \sqrt [3]{-x^3} x \Gamma \left (-\frac {1}{3},-\frac {x^3}{3}\right )+3 \sqrt [3]{3} c_1 \left (-x^3\right )^{2/3} \Gamma \left (\frac {1}{3},-\frac {x^3}{3}\right )+18 c_2 x^2\right )}{18 x} \]