Internal problem ID [9891]
Internal file name [OUTPUT/8838_Monday_June_06_2022_05_36_10_AM_65260830/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_high_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+\left (6-4 a \right ) x^{3} y^{\prime \prime \prime }+\left (4 b^{2} c^{2} x^{2 c}+6 \left (a -1\right )^{2}-2 c^{2} \left (\mu ^{2}+\nu ^{2}\right )+1\right ) x^{2} y^{\prime \prime }+\left (4 \left (3 c -2 a +1\right ) b^{2} c^{2} x^{2 c}+\left (2 a -1\right ) \left (2 c^{2} \left (\mu ^{2}+\nu ^{2}\right )-2 a \left (a -1\right )-1\right )\right ) x y^{\prime }+\left (4 \left (a -c \right ) \left (a -2 c \right ) b^{2} c^{2} x^{2 c}+\left (c \mu +c \nu +a \right ) \left (c \mu +c \nu -a \right ) \left (c \mu -c \nu +a \right ) \left (c \mu -c \nu -a \right )\right ) y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{4} \left (\frac {d}{d x}\frac {d}{d x}y^{\prime \prime }\right )+\left (6-4 a \right ) x^{3} \left (\frac {d}{d x}y^{\prime \prime }\right )+\left (4 b^{2} c^{2} x^{2 c}+6 \left (a -1\right )^{2}-2 c^{2} \left (\mu ^{2}+\nu ^{2}\right )+1\right ) x^{2} \left (\frac {d}{d x}y^{\prime }\right )+\left (4 \left (3 c -2 a +1\right ) b^{2} c^{2} x^{2 c}+\left (2 a -1\right ) \left (2 c^{2} \left (\mu ^{2}+\nu ^{2}\right )-2 a \left (a -1\right )-1\right )\right ) x y^{\prime }+\left (4 \left (a -c \right ) \left (a -2 c \right ) b^{2} c^{2} x^{2 c}+\left (c \mu +c \nu +a \right ) \left (c \mu +c \nu -a \right ) \left (c \mu -c \nu +a \right ) \left (c \mu -c \nu -a \right )\right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime } \end {array} \]
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying high order exact linear fully integrable trying to convert to a linear ODE with constant coefficients trying differential order: 4; missing the dependent variable trying a solution in terms of MeijerG functions -> Try computing a Rational Normal Form for the given ODE... RNF transformation found: {t = x^(2*c), u(t) = x^(-3/2+3*c)*y(x)*exp(-Int(1/2/x*(-3+2*a),x))} -> Calling odsolve with the ODE`, diff(diff(diff(diff(u(t), t), t), t), t) = (-(1/2)*b^2/t^3+(-(1/16)*mu^4+(1/8)*mu^2*nu^2-(1/16)*nu Methods for high order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying high order exact linear fully integrable trying to convert to a linear ODE with constant coefficients trying differential order: 4; missing the dependent variable Multiplying solutions by`, exp(Int(-(1/2)/t, t))` Equation is the symmetric product of`, diff(diff(y(t), t), t)+(1/4)*(b^2*t-nu checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful <- solving the Rational Normal Form successful`
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 63
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)
\[ y \left (x \right ) = x^{a} \left (\left (\operatorname {BesselJ}\left (\mu , b \,x^{c}\right ) c_{1} +\operatorname {BesselY}\left (\mu , b \,x^{c}\right ) c_{3} \right ) \operatorname {BesselJ}\left (\nu , b \,x^{c}\right )+\operatorname {BesselY}\left (\nu , b \,x^{c}\right ) \left (c_{4} \operatorname {BesselY}\left (\mu , b \,x^{c}\right )+\operatorname {BesselJ}\left (\mu , b \,x^{c}\right ) c_{2} \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.102 (sec). Leaf size: 304
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]
\[ y(x)\to b^{\frac {a-c (\mu +\nu )}{c}} \left (x^{2 c}\right )^{\frac {a-c (\mu +\nu )}{2 c}} \left (c_1 \, _2F_3\left (-\frac {\mu }{2}-\frac {\nu }{2}+\frac {1}{2},-\frac {\mu }{2}-\frac {\nu }{2}+1;1-\mu ,1-\nu ,-\mu -\nu +1;-b^2 x^{2 c}\right )+c_2 b^{2 \mu } \left (x^{2 c}\right )^{\mu } \, _2F_3\left (\frac {\mu }{2}-\frac {\nu }{2}+\frac {1}{2},\frac {\mu }{2}-\frac {\nu }{2}+1;\mu +1,1-\nu ,\mu -\nu +1;-b^2 x^{2 c}\right )+b^{2 \nu } \left (x^{2 c}\right )^{\nu } \left (c_3 \, _2F_3\left (-\frac {\mu }{2}+\frac {\nu }{2}+\frac {1}{2},-\frac {\mu }{2}+\frac {\nu }{2}+1;1-\mu ,\nu +1,-\mu +\nu +1;-b^2 x^{2 c}\right )+c_4 b^{2 \mu } \left (x^{2 c}\right )^{\mu } \, _2F_3\left (\frac {\mu }{2}+\frac {\nu }{2}+\frac {1}{2},\frac {\mu }{2}+\frac {\nu }{2}+1;\mu +1,\nu +1,\mu +\nu +1;-b^2 x^{2 c}\right )\right )\right ) \]