Internal problem ID [6027]
Internal file name [OUTPUT/5275_Sunday_June_05_2022_03_29_13_PM_38106072/index.tex]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 3. Linear equations with variable coefficients. Page 130
Problem number: 5.
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.
\[ \boxed {y^{\prime \prime \prime }-x y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 0] \end {align*}
Unable to solve this ODE. Initial conditions are used to solve for the constants of integration.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d x}y^{\prime \prime }-x y=0, y \left (0\right )=1, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=0, \left (\frac {d}{d x}y^{\prime }\right )\bigg | {\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=0\right ] \\ \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 \cdot y\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x \cdot y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -1 \\ {} & {} & x \cdot y=\moverset {\infty }{\munderset {k =1}{\sum }}a_{k -1} 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 (\moverset {\infty }{\munderset {k =1}{\sum }}\left (a_{k +3} \left (k +3\right ) \left (k +2\right ) \left (k +1\right )-a_{k -1}\right ) x^{k}\right )=0 \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & 6 a_{3}=0 \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (k^{3}+6 k^{2}+11 k +6\right ) a_{k +3}-a_{k -1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & \left (\left (k +1\right )^{3}+6 \left (k +1\right )^{2}+11 k +17\right ) a_{k +4}-a_{k}=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 +4}=\frac {a_{k}}{k^{3}+9 k^{2}+26 k +24}, 6 a_{3}=0\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 successful: received ODE is equivalent to the 0F2 ODE, case c = 0 `
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 14
dsolve([diff(y(x),x$3)-x*y(x)=0,y(0) = 1, D(y)(0) = 0, (D@@2)(y)(0) = 0],y(x), singsol=all)
\[ y \left (x \right ) = \operatorname {hypergeom}\left (\left [\right ], \left [\frac {1}{2}, \frac {3}{4}\right ], \frac {x^{4}}{64}\right ) \]
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 21
DSolve[{y'''[x]-x*y[x]==0,{y[0]==1,y'[0]==0,y''[0]==0}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \, _0F_2\left (;\frac {1}{2},\frac {3}{4};\frac {x^4}{64}\right ) \]