Internal problem ID [9908]
Internal file name [OUTPUT/8855_Monday_June_06_2022_05_40_55_AM_71490738/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 5, linear fifth and higher order
Problem number: 1586.
ODE order: 5.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_high_order, _missing_y]]
Unable to solve or complete the solution.
\[ \boxed {x y^{\left (5\right )}-\left (\left (a A_{2} -A_{1} \right ) x +A_{2} \right ) y^{\prime }=\left (a A_{1} -A_{0} \right ) x +A_{1}} \] Since \(y\) is missing from the ode then we can use the substitution \(y^{\prime } = v \left (x \right )\) to reduce the order by one. The ODE becomes \begin {align*} \left (-A_{2} a x +A_{1} x -A_{2} \right ) v \left (x \right )+v^{\prime \prime \prime \prime }\left (x \right ) x = 0 \end {align*}
Unable to solve this ODE. Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}\frac {d^{2}}{d x^{2}}y^{\prime \prime }\right ) x -\left (\left (a A_{2} -A_{1} \right ) x +A_{2} \right ) y^{\prime }=\left (a A_{1} -A_{0} \right ) x +A_{1} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 5 \\ {} & {} & \frac {d}{d x}\frac {d^{2}}{d x^{2}}y^{\prime \prime } \end {array} \]
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 5; linear nonhomogeneous with symmetry [0,1] -> Calling odsolve with the ODE`, diff(diff(diff(diff(_b(_a), _a), _a), _a), _a) = -(-A__2*_b(_a)*a*_a+A__1*_b(_a)*_a-A__1*a*_a+A__0 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 Multiplying solutions by`, exp(Int(-(1/2)/_a, _a))` Multiplying solutions by`, exp(Int(-1/_a, _a))` trying a solution in term trying reduction of order using simple exponentials --- Trying Lie symmetry methods, high order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 5 <- differential order: 5; linear nonhomogeneous with symmetry [0,1] successful`
✗ Solution by Maple
dsolve(x*diff(y(x),x$5)-((a*A__1-A__0)*x+A__1)-((a*A__2-A__1)*x+A__2)*diff(y(x),x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[x*y'''''[x]-((a*A1-A0)*x+A1)-((a*A2-A1)*x+A2)*y'[x]== 0,y[x],x,IncludeSingularSolutions -> True]
Not solved