Internal problem ID [9140]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1565.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _with_linear_symmetries]]
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+6 y^{\prime \prime \prime } x^{3}+\left (4 x^{4}+\left (-\rho ^{2}-\sigma ^{2}+7\right ) x^{2}\right ) y^{\prime \prime }+\left (16 x^{3}+\left (-\rho ^{2}-\sigma ^{2}+1\right ) x \right ) y^{\prime }+\left (\rho ^{2} \sigma ^{2}+8 x^{2}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 85
dsolve(x^4*diff(diff(diff(diff(y(x),x),x),x),x)+6*x^3*diff(diff(diff(y(x),x),x),x)+(4*x^4+(-rho^2-sigma^2+7)*x^2)*diff(diff(y(x),x),x)+(16*x^3+(-rho^2-sigma^2+1)*x)*diff(y(x),x)+(rho^2*sigma^2+8*x^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {BesselJ}\left (\frac {\sigma }{2}+\frac {\rho }{2}, x\right ) \operatorname {BesselJ}\left (-\frac {\sigma }{2}+\frac {\rho }{2}, x\right )+c_{2} \operatorname {BesselJ}\left (\frac {\sigma }{2}+\frac {\rho }{2}, x\right ) \operatorname {BesselY}\left (-\frac {\sigma }{2}+\frac {\rho }{2}, x\right )+c_{3} \operatorname {BesselY}\left (\frac {\sigma }{2}+\frac {\rho }{2}, x\right ) \operatorname {BesselJ}\left (-\frac {\sigma }{2}+\frac {\rho }{2}, x\right )+c_{4} \operatorname {BesselY}\left (\frac {\sigma }{2}+\frac {\rho }{2}, x\right ) \operatorname {BesselY}\left (-\frac {\sigma }{2}+\frac {\rho }{2}, x\right ) \]
✓ Solution by Mathematica
Time used: 0.273 (sec). Leaf size: 174
DSolve[(rho^2*sigma^2 + 8*x^2)*y[x] + ((1 - rho^2 - sigma^2)*x + 16*x^3)*y'[x] + ((7 - rho^2 - sigma^2)*x^2 + 4*x^4)*y''[x] + 6*x^3*Derivative[3][y][x] + x^4*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 2^{-\sigma } \operatorname {Gamma}\left (\frac {1}{2} (\rho -\sigma +2)\right ) \operatorname {BesselJ}\left (\frac {\rho -\sigma }{2},x\right ) \left (c_3 \operatorname {Gamma}\left (-\frac {\rho }{2}-\frac {\sigma }{2}+1\right ) \operatorname {BesselJ}\left (\frac {1}{2} (-\rho -\sigma ),x\right )+c_2 2^{\rho +\sigma } \operatorname {Gamma}\left (\frac {1}{2} (\rho +\sigma +2)\right ) \operatorname {BesselJ}\left (\frac {\rho +\sigma }{2},x\right )\right )+2^{-\rho } \operatorname {Gamma}\left (\frac {1}{2} (-\rho +\sigma +2)\right ) \operatorname {BesselJ}\left (\frac {\sigma -\rho }{2},x\right ) \left (c_1 \operatorname {Gamma}\left (-\frac {\rho }{2}-\frac {\sigma }{2}+1\right ) \operatorname {BesselJ}\left (\frac {1}{2} (-\rho -\sigma ),x\right )+c_4 2^{\rho +\sigma } \operatorname {Gamma}\left (\frac {1}{2} (\rho +\sigma +2)\right ) \operatorname {BesselJ}\left (\frac {\rho +\sigma }{2},x\right )\right ) \\ \end{align*}