Internal problem ID [9874]
Internal file name [OUTPUT/8821_Monday_June_06_2022_05_34_05_AM_72377141/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1552.
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, _linear, _nonhomogeneous]]
Unable to solve or complete the solution.
\[ \boxed {x^{2} y^{\prime \prime \prime \prime }+2 x y^{\prime \prime \prime }+a y=b \,x^{2}} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d}{d x}\frac {d}{d x}y^{\prime \prime }\right )+2 \left (\frac {d}{d x}y^{\prime \prime }\right ) x +a y=b \,x^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime } \end {array} \]
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 4; linear nonhomogeneous with symmetry [0,1] trying high order linear exact nonhomogeneous trying differential order: 4; missing the dependent variable checking if the LODE is of Euler type Equation is the LCLM of (-a)^(1/2)/x*y(x)+diff(diff(y(x),x),x), -(-a)^(1/2)/x*y(x)+diff(diff(y(x),x),x) 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 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 LCLM ode successful `
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 94
dsolve(x^2*diff(y(x),x$4)+2*x*diff(y(x),x$3)+a*y(x)-b*x^2=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{4} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {-\sqrt {-a}}\, \sqrt {x}\right ) a +c_{3} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {-\sqrt {-a}}\, \sqrt {x}\right ) a +c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \left (-a \right )^{\frac {1}{4}} \sqrt {x}\right ) a +c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \left (-a \right )^{\frac {1}{4}} \sqrt {x}\right ) a +b \,x^{2}}{a} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[x^2*y''''[x]+2*x*y'''[x]+a*y[x]-b*x^2==0,y[x],x,IncludeSingularSolutions -> True]
Timed out