Internal problem ID [9148]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1569.
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 (-4 a +6\right ) x^{3} y^{\prime \prime \prime }+\left (4 b^{2} c^{2} x^{2 c}+6 \left (a -1\right )^{2}-2 c^{2} \left (\mu ^{2}+\nu ^{2}\right )+1\right ) x^{2} y^{\prime \prime }+\left (4 \left (3 c -2 a +1\right ) b^{2} c^{2} x^{2 c}+\left (-1+2 a \right ) \left (2 c^{2} \left (\mu ^{2}+\nu ^{2}\right )-2 a \left (a -1\right )-1\right )\right ) x y^{\prime }+\left (4 \left (a -c \right ) \left (a -2 c \right ) b^{2} c^{2} x^{2 c}+\left (c \mu +c \nu +a \right ) \left (c \mu +c \nu -a \right ) \left (c \mu -c \nu +a \right ) \left (c \mu -c \nu -a \right )\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 81
dsolve(x^4*diff(diff(diff(diff(y(x),x),x),x),x)+(6-4*a)*x^3*diff(diff(diff(y(x),x),x),x)+(4*b^2*c^2*x^(2*c)+6*(a-1)^2-2*c^2*(mu^2+nu^2)+1)*x^2*diff(diff(y(x),x),x)+(4*(3*c-2*a+1)*b^2*c^2*x^(2*c)+(2*a-1)*(2*c^2*(mu^2+nu^2)-2*a*(a-1)-1))*x*diff(y(x),x)+(4*(a-c)*(a-2*c)*b^2*c^2*x^(2*c)+(c*mu+c*nu+a)*(c*mu+c*nu-a)*(c*mu-c*nu+a)*(c*mu-c*nu-a))*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{a} \BesselJ \left (\mu , b \,x^{c}\right ) \BesselY \left (\nu , b \,x^{c}\right )+c_{2} x^{a} \BesselJ \left (\nu , b \,x^{c}\right ) \BesselJ \left (\mu , b \,x^{c}\right )+c_{3} x^{a} \BesselJ \left (\nu , b \,x^{c}\right ) \BesselY \left (\mu , b \,x^{c}\right )+c_{4} x^{a} \BesselY \left (\nu , b \,x^{c}\right ) \BesselY \left (\mu , b \,x^{c}\right ) \]
✓ Solution by Mathematica
Time used: 0.066 (sec). Leaf size: 230
DSolve[x^4*y''''[x]+(6-4*a)*x^3*y'''[x]+(4*b^2*c^2*x^(2*c)+6*(a-1)^2-2*c^2*(\[Mu]^2+\[Nu]^2)+1)*x^2*y''[x]+(4*(3*c-2*a+1)*b^2*c^2*x^(2*c)+(2*a-1)*(2*c^2*(\[Mu]^2+\[Nu]^2)-2*a*(a-1)-1))*x*y'[x]+(4*(a-c)*(a-2*c)*b^2*c^2*x^(2*c)+(c*\[Mu]+c*\[Nu]+a)*(c*\[Mu]+c*\[Nu]-a)*(c*\[Mu]-c*\[Nu]+a)*(c*\[Mu]-c*\[Nu]-a))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 2^{-\mu -\nu } b^{\frac {a-c (\mu +\nu )}{c}} \left (x^{2 c}\right )^{\frac {a-c (\mu +\nu )}{2 c}} \left (b x^c\right )^{-\mu -\nu } \left (4^{\mu } b^{2 \mu } \Gamma (\mu +1) \left (x^{2 c}\right )^{\mu } J_{\mu }\left (b x^c\right ) \left (c_4 4^{\nu } b^{2 \nu } \Gamma (\nu +1) \left (x^{2 c}\right )^{\nu } J_{\nu }\left (b x^c\right )+c_2 \Gamma (1-\nu ) \left (b x^c\right )^{2 \nu } J_{-\nu }\left (b x^c\right )\right )+\Gamma (1-\mu ) \left (b x^c\right )^{2 \mu } J_{-\mu }\left (b x^c\right ) \left (c_3 4^{\nu } b^{2 \nu } \Gamma (\nu +1) \left (x^{2 c}\right )^{\nu } J_{\nu }\left (b x^c\right )+c_1 \Gamma (1-\nu ) \left (b x^c\right )^{2 \nu } J_{-\nu }\left (b x^c\right )\right )\right ) \\ \end{align*}