Internal problem ID [6150]
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: 0.088 (sec). Leaf size: 56
DSolve[y'''[x]+x^2*y''[x]+5*x*y'[x]+3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{6} e^{-\frac {x^3}{3}} \left (2 c_3 \operatorname {ExpIntegralE}\left (\frac {4}{3},-\frac {x^3}{3}\right )+c_1 x^2 \operatorname {ExpIntegralE}\left (\frac {2}{3},-\frac {x^3}{3}\right )-6 c_2 x\right ) \\ \end{align*}