Internal problem ID [9149]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1570.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{4} y^{\prime \prime \prime \prime }+\left (6-4 a -4 c \right ) x^{3} y^{\prime \prime \prime }+\left (-2 \nu ^{2} c^{2}+2 a^{2}+4 \left (c -1+a \right )^{2}+4 \left (-1+a \right ) \left (-1+c \right )-1\right ) x^{2} y^{\prime \prime }+\left (2 \nu ^{2} c^{2}-2 a^{2}-\left (-1+2 a \right ) \left (2 c -1\right )\right ) \left (2 a +2 c -1\right ) x y^{\prime }+\left (\left (-\nu ^{2} c^{2}+a^{2}\right ) \left (-\nu ^{2} c^{2}+a^{2}+4 c a +4 c^{2}\right )-b^{4} c^{4} x^{4 c}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.026 (sec). Leaf size: 57
dsolve(x^4*diff(diff(diff(diff(y(x),x),x),x),x)+(6-4*a-4*c)*x^3*diff(diff(diff(y(x),x),x),x)+(-2*nu^2*c^2+2*a^2+4*(a+c-1)^2+4*(a-1)*(c-1)-1)*x^2*diff(diff(y(x),x),x)+(2*nu^2*c^2-2*a^2-(2*a-1)*(2*c-1))*(2*a+2*c-1)*x*diff(y(x),x)+((-c^2*nu^2+a^2)*(-c^2*nu^2+a^2+4*a*c+4*c^2)-b^4*c^4*x^(4*c))*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{a} \BesselJ \left (\nu , b \,x^{c}\right )+c_{2} x^{a} \BesselY \left (\nu , b \,x^{c}\right )+c_{3} x^{a} \BesselJ \left (\nu , i b \,x^{c}\right )+c_{4} x^{a} \BesselY \left (\nu , i b \,x^{c}\right ) \]
✓ Solution by Mathematica
Time used: 0.074 (sec). Leaf size: 203
DSolve[((a^2 - c^2*\[Nu]^2)*(a^2 + 4*a*c + 4*c^2 - c^2*\[Nu]^2) - b^4*c^4*x^(4*c))*y[x] + (-1 + 2*a + 2*c)*(-2*a^2 - (-1 + 2*a)*(-1 + 2*c) + 2*c^2*\[Nu]^2)*x*y'[x] + (-1 + 2*a^2 + 4*(-1 + a)*(-1 + c) + 4*(-1 + a + c)^2 - 2*c^2*\[Nu]^2)*x^2*y''[x] + (6 - 4*a - 4*c)*x^3*Derivative[3][y][x] + x^4*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to b^{a/c} (-1)^{\frac {a-c \nu }{4 c}} 2^{-\frac {2 a}{c}-\nu -3} \left (x^{4 c}\right )^{\frac {a}{4 c}} \left (4^{\nu } (i c_2 (\nu -1)+4 c_1) \Gamma (1-\nu ) J_{-\nu }\left (b \sqrt [4]{x^{4 c}}\right )+4^{\nu } (4 c_1-i c_2 (\nu -1)) \Gamma (1-\nu ) I_{-\nu }\left (b \sqrt [4]{x^{4 c}}\right )+i^{\nu } \Gamma (\nu +1) \left ((i c_4 (\nu +1)+4 c_3) I_{\nu }\left (b \sqrt [4]{x^{4 c}}\right )-i (c_4 \nu +4 i c_3+c_4) J_{\nu }\left (b \sqrt [4]{x^{4 c}}\right )\right )\right ) \\ \end{align*}