Internal problem ID [5817]
Internal file name [OUTPUT/5065_Sunday_June_05_2022_03_19_53_PM_91565691/index.tex]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 2. Linear homogeneous equations. Section 2.2 problems. page 95
Problem number: 56.
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.
\[ \boxed {y^{\prime \prime \prime }-2 x y^{\prime \prime }+4 y^{\prime } x^{2}+8 x^{3} y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }-2 \left (\frac {d}{d x}y^{\prime }\right ) x +4 y^{\prime } x^{2}+8 x^{3} y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime } \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \\ {} & {} & y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{3}\cdot y\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{3}\cdot y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +3} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -3 \\ {} & {} & x^{3}\cdot y=\moverset {\infty }{\munderset {k =3}{\sum }}a_{k -3} x^{k} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{2}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{2}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} k \,x^{k +1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -1 \\ {} & {} & x^{2}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =1}{\sum }}a_{k -1} \left (k -1\right ) x^{k} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x \cdot \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x \cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =1}{\sum }}a_{k} k \left (k -1\right ) x^{k -1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & x \cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k +1} \left (k +1\right ) k \,x^{k} \\ {} & \circ & \textrm {Convert}\hspace {3pt} \frac {d}{d x}y^{\prime \prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }=\moverset {\infty }{\munderset {k =3}{\sum }}a_{k} k \left (k -1\right ) \left (k -2\right ) x^{k -3} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k +3} \left (k +3\right ) \left (k +2\right ) \left (k +1\right ) x^{k} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & 6 a_{3}+\left (24 a_{4}-4 a_{2}\right ) x +\left (60 a_{5}-12 a_{3}+4 a_{1}\right ) x^{2}+\left (\moverset {\infty }{\munderset {k =3}{\sum }}\left (a_{k +3} \left (k +3\right ) \left (k +2\right ) \left (k +1\right )-2 a_{k +1} \left (k +1\right ) k +4 a_{k -1} \left (k -1\right )+8 a_{k -3}\right ) x^{k}\right )=0 \\ \bullet & {} & \textrm {The coefficients of each power of}\hspace {3pt} x \hspace {3pt}\textrm {must be 0}\hspace {3pt} \\ {} & {} & \left [6 a_{3}=0, 24 a_{4}-4 a_{2}=0, 60 a_{5}-12 a_{3}+4 a_{1}=0\right ] \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & \left \{a_{3}=0, a_{4}=\frac {a_{2}}{6}, a_{5}=-\frac {a_{1}}{15}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & k^{3} a_{k +3}+\left (-2 a_{k +1}+6 a_{k +3}\right ) k^{2}+\left (4 a_{k -1}-2 a_{k +1}+11 a_{k +3}\right ) k +8 a_{k -3}-4 a_{k -1}+6 a_{k +3}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +3 \\ {} & {} & \left (k +3\right )^{3} a_{k +6}+\left (-2 a_{k +4}+6 a_{k +6}\right ) \left (k +3\right )^{2}+\left (4 a_{k +2}-2 a_{k +4}+11 a_{k +6}\right ) \left (k +3\right )+8 a_{k}-4 a_{k +2}+6 a_{k +6}=0 \\ \bullet & {} & \textrm {Recursion relation that defines the series solution to the ODE}\hspace {3pt} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k}, a_{k +6}=\frac {2 \left (k^{2} a_{k +4}-2 k a_{k +2}+7 k a_{k +4}-4 a_{k}-4 a_{k +2}+12 a_{k +4}\right )}{k^{3}+15 k^{2}+74 k +120}, a_{3}=0, a_{4}=\frac {a_{2}}{6}, a_{5}=-\frac {a_{1}}{15}\right ] \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 trying Louvillian solutions for 3rd order ODEs, imprimitive case -> pFq: Equivalence to the 3F2 or one of its 3 confluent cases under a power @ Moebius -> pFq: Equivalence to the 3F2 or one of its 3 confluent cases under a power @ Moebius trying a solution in terms of MeijerG functions -> pFq: Equivalence to the 3F2 or one of its 3 confluent cases under a power @ Moebius -> pFq: Equivalence to the 3F2 or one of its 3 confluent cases under a power @ Moebius trying a solution in terms of MeijerG functions checking if the LODE is of Euler type <- no solution through differential factorization was found trying reduction of order using simple exponentials --- Trying Lie symmetry methods, high order --- `, `-> Computing symmetries using: way = 3`[0, y]
✗ Solution by Maple
dsolve(diff(y(x),x$3)-2*x*diff(y(x),x$2)+4*x^2*diff(y(x),x)+8*x^3*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'''[x]-2*x*y''[x]+4*x^2*y'[x]+8*x^3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved