Internal problem ID [7468]
Book: Second order enumerated odes
Section: section 2
Problem number: 27.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-x^{2} y^{\prime }+y x=x^{m +1}} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 201
dsolve(diff(y(x),x$2)-x^2*diff(y(x),x)+x*y(x)=x^(m+1),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-3 \,3^{\frac {m}{6}} {\mathrm e}^{\frac {x^{3}}{6}} \left (x^{3}\right )^{-\frac {m}{6}} \operatorname {WhittakerM}\left (\frac {m}{6}, \frac {m}{6}+\frac {1}{2}, \frac {x^{3}}{3}\right ) x^{m}+\left (m +3\right ) \left (3^{\frac {1}{3}} {\mathrm e}^{\frac {x^{3}}{3}} c_{1} -\frac {\left (\int \frac {\left (-3 \left (-x^{3}\right )^{\frac {2}{3}}+x^{3} 3^{\frac {2}{3}} {\mathrm e}^{-\frac {x^{3}}{3}} \left (\Gamma \left (\frac {2}{3}\right )-\Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right )\right )\right ) x^{m +1}}{\left (-x^{3}\right )^{\frac {2}{3}}}d x -3 c_{2} \right ) x}{3}\right )\right ) \left (-x^{3}\right )^{\frac {2}{3}}-\frac {\left (\left (x^{3}\right )^{-\frac {m}{6}} x^{m +3} 3^{\frac {5}{3}+\frac {m}{6}} {\mathrm e}^{-\frac {x^{3}}{6}} \operatorname {WhittakerM}\left (\frac {m}{6}, \frac {m}{6}+\frac {1}{2}, \frac {x^{3}}{3}\right )-3 c_{1} x^{3} \left (m +3\right )\right ) \left (\Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right )-\Gamma \left (\frac {2}{3}\right )\right )}{3}}{\left (-x^{3}\right )^{\frac {2}{3}} \left (m +3\right )} \]
✓ Solution by Mathematica
Time used: 0.453 (sec). Leaf size: 144
DSolve[y''[x]-x^2*y'[x]+x*y[x]==x^(m+1),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x \int _1^x\frac {e^{-\frac {1}{3} K[1]^3} \Gamma \left (-\frac {1}{3},-\frac {1}{3} K[1]^3\right ) K[1]^{m+1} \sqrt [3]{-K[1]^3}}{3 \sqrt [3]{3}}dK[1]-\frac {\sqrt [3]{-x^3} \left (x^3\right )^{-m/3} \Gamma \left (-\frac {1}{3},-\frac {x^3}{3}\right ) \left (-3^{m/3} x^m \Gamma \left (\frac {m+3}{3},\frac {x^3}{3}\right )+c_2 \left (x^3\right )^{m/3}\right )}{3 \sqrt [3]{3}}+c_1 x \]