Internal problem ID [9834]
Internal file name [OUTPUT/8779_Monday_June_06_2022_05_26_57_AM_27955589/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1510.
ODE order: 3.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_3rd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {x^{3} y^{\prime \prime \prime }+\left (a \,x^{2 \nu }+1-\nu ^{2}\right ) x y^{\prime }+\left (b \,x^{3 \nu }+a \left (-1+\nu \right ) x^{2 \nu }+\nu ^{2}-1\right ) y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{3} \left (\frac {d}{d x}y^{\prime \prime }\right )+\left (a \,x^{2 \nu }+1-\nu ^{2}\right ) x y^{\prime }+\left (b \,x^{3 \nu }+a \left (-1+\nu \right ) x^{2 \nu }+\nu ^{2}-1\right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime } \end {array} \]
Maple trace
`Methods for third 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: 3; missing the dependent variable -> pFq: Equivalence to the 3F2 or one of its 3 confluent cases under a power @ Moebius trying a solution in terms of MeijerG functions trying reduction of order using simple exponentials trying differential order: 3; exact nonlinear --- Trying Lie symmetry methods, high order --- `, `-> Computing symmetries using: way = 3`[0, y]
✗ Solution by Maple
dsolve(x^3*diff(diff(diff(y(x),x),x),x)+(a*x^(2*nu)+1-nu^2)*x*diff(y(x),x)+(b*x^(3*nu)+a*(nu-1)*x^(2*nu)+nu^2-1)*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.022 (sec). Leaf size: 102
DSolve[(-1 + nu^2 + a*(-1 + nu)*x^(2*nu) + b*x^(3*nu))*y[x] + x*(1 - nu^2 + a*x^(2*nu))*y'[x] + x^3*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 x^{1-\nu } e^{\frac {x^{\nu } \text {Root}\left [\text {$\#$1}^3+\text {$\#$1} a+b\&,1\right ]}{\nu }}+c_2 x^{1-\nu } e^{\frac {x^{\nu } \text {Root}\left [\text {$\#$1}^3+\text {$\#$1} a+b\&,2\right ]}{\nu }}+c_3 x^{1-\nu } e^{\frac {x^{\nu } \text {Root}\left [\text {$\#$1}^3+\text {$\#$1} a+b\&,3\right ]}{\nu }} \]