Internal problem ID [9150]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1571.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\nu ^{4} x^{4} y^{\prime \prime \prime \prime }+\left (4 \nu -2\right ) \nu ^{3} x^{3} y^{\prime \prime \prime }+\left (\nu -1\right ) \left (2 \nu -1\right ) \nu ^{2} x^{2} y^{\prime \prime }-\frac {b^{4} x^{\frac {2}{\nu }} y}{16}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.092 (sec). Leaf size: 151
dsolve(nu^4*x^4*diff(diff(diff(diff(y(x),x),x),x),x)+(4*nu-2)*nu^3*x^3*diff(diff(diff(y(x),x),x),x)+(nu-1)*(2*nu-1)*nu^2*x^2*diff(diff(y(x),x),x)-1/16*b^4*x^(2/nu)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {x}\, \BesselJ \left (\frac {1}{{\lfloor \frac {1}{\nu }\rfloor }}, \frac {\sqrt {\frac {b^{2}}{\nu ^{2}}}\, x^{\frac {{\lfloor \frac {1}{\nu }\rfloor }}{2}}}{{\lfloor \frac {1}{\nu }\rfloor }}\right )+c_{2} \sqrt {x}\, \BesselY \left (\frac {1}{{\lfloor \frac {1}{\nu }\rfloor }}, \frac {\sqrt {\frac {b^{2}}{\nu ^{2}}}\, x^{\frac {{\lfloor \frac {1}{\nu }\rfloor }}{2}}}{{\lfloor \frac {1}{\nu }\rfloor }}\right )+c_{3} \sqrt {x}\, \BesselJ \left (\frac {1}{{\lfloor \frac {1}{\nu }\rfloor }}, \frac {\sqrt {-\frac {b^{2}}{\nu ^{2}}}\, x^{\frac {{\lfloor \frac {1}{\nu }\rfloor }}{2}}}{{\lfloor \frac {1}{\nu }\rfloor }}\right )+c_{4} \sqrt {x}\, \BesselY \left (\frac {1}{{\lfloor \frac {1}{\nu }\rfloor }}, \frac {\sqrt {-\frac {b^{2}}{\nu ^{2}}}\, x^{\frac {{\lfloor \frac {1}{\nu }\rfloor }}{2}}}{{\lfloor \frac {1}{\nu }\rfloor }}\right ) \]
✓ Solution by Mathematica
Time used: 0.045 (sec). Leaf size: 185
DSolve[-1/16*(b^4*x^(2/\[Nu])*y[x]) + (-1 + \[Nu])*\[Nu]^2*(-1 + 2*\[Nu])*x^2*y''[x] + \[Nu]^3*(-2 + 4*\[Nu])*x^3*y'''[x] + \[Nu]^4*x^4*y''''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 8^{-\nu -1} b^{\nu } \left (x^{2/\nu }\right )^{\nu /4} \left (4^{\nu } (i c_2 (\nu -1)+4 c_1) \Gamma (1-\nu ) J_{-\nu }\left (b \sqrt [4]{x^{2/\nu }}\right )+4^{\nu } (4 c_1-i c_2 (\nu -1)) \Gamma (1-\nu ) I_{-\nu }\left (b \sqrt [4]{x^{2/\nu }}\right )+i^{\nu } \Gamma (\nu +1) \left ((i c_4 (\nu +1)+4 c_3) I_{\nu }\left (b \sqrt [4]{x^{2/\nu }}\right )-i (c_4 \nu +4 i c_3+c_4) J_{\nu }\left (b \sqrt [4]{x^{2/\nu }}\right )\right )\right ) \\ \end{align*}