Internal problem ID [9129]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1550.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x y^{\prime \prime \prime \prime }-\left (6 x^{2}+1\right ) y^{\prime \prime \prime }+12 y^{\prime \prime } x^{3}-\left (9 x^{2}-7\right ) x^{2} y^{\prime }+2 \left (x^{2}-3\right ) x^{3} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 159
dsolve(x*diff(diff(diff(diff(y(x),x),x),x),x)-(6*x^2+1)*diff(diff(diff(y(x),x),x),x)+12*x^3*diff(diff(y(x),x),x)-(9*x^2-7)*x^2*diff(y(x),x)+2*(x^2-3)*x^3*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{x^{2}}+c_{2} {\mathrm e}^{\frac {x^{2}}{2}}+c_{3} \left (-{\mathrm e}^{x^{2}} \left (\int \frac {\WhittakerM \left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {x^{2}}{4}}}{x^{\frac {3}{2}}}d x \right )+\left (\int \frac {\WhittakerM \left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{x^{\frac {3}{2}}}d x \right ) {\mathrm e}^{\frac {x^{2}}{2}}\right )+c_{4} \left (-{\mathrm e}^{x^{2}} \left (\int \frac {\WhittakerW \left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {x^{2}}{4}}}{x^{\frac {3}{2}}}d x \right )+\left (\int \frac {\WhittakerW \left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{x^{\frac {3}{2}}}d x \right ) {\mathrm e}^{\frac {x^{2}}{2}}\right ) \]
✓ Solution by Mathematica
Time used: 2.073 (sec). Leaf size: 213
DSolve[2*x^3*(-3 + x^2)*y[x] - x^2*(-7 + 9*x^2)*y'[x] + 12*x^3*y''[x] - (1 + 6*x^2)*Derivative[3][y][x] + x*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{\frac {x^2}{2}} \left (c_3 \int _1^x\frac {e^{\frac {K[1]^2}{2}} \left (\int \frac {e^{\frac {1}{4} \left (-1+\sqrt {5}\right ) K[1]^2} \text {HypergeometricU}\left (-\frac {1}{4}+\frac {9}{4 \sqrt {5}},-\frac {1}{2},-\frac {1}{2} \sqrt {5} K[1]^2\right ) \left (K[1]^2\right )^{3/4}}{K[1]^{7/2}} \, dK[1]\right ) K[1]}{\sqrt [4]{2}}dK[1]+c_4 \int _1^x\frac {e^{\frac {K[2]^2}{2}} \left (\int \frac {e^{\frac {1}{4} \left (-1+\sqrt {5}\right ) K[2]^2} \left (K[2]^2\right )^{3/4} L_{\frac {1}{4}-\frac {9}{4 \sqrt {5}}}^{-\frac {3}{2}}\left (-\frac {1}{2} \sqrt {5} K[2]^2\right )}{K[2]^{7/2}} \, dK[2]\right ) K[2]}{\sqrt [4]{2}}dK[2]+c_1\right )+c_2 e^{x^2} \\ \end{align*}