Internal problem ID [9795]
Internal file name [OUTPUT/8738_Monday_June_06_2022_05_22_11_AM_4020438/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1469.
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.
\[ \boxed {y^{\prime \prime \prime }+3 a x y^{\prime \prime }+3 a^{2} x^{2} y^{\prime }+a^{3} x^{3} y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }+3 a x \left (\frac {d}{d x}y^{\prime }\right )+3 a^{2} x^{2} y^{\prime }+a^{3} 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 (6 a_{2} a +24 a_{4}\right ) x +\left (3 a_{1} a^{2}+18 a a_{3}+60 a_{5}\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 )+3 a a_{k +1} \left (k +1\right ) k +3 a^{2} a_{k -1} \left (k -1\right )+a^{3} 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, 6 a_{2} a +24 a_{4}=0, 3 a_{1} a^{2}+18 a a_{3}+60 a_{5}=0\right ] \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & \left \{a_{3}=0, a_{4}=-\frac {a_{2} a}{4}, a_{5}=-\frac {a_{1} a^{2}}{20}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & k^{3} a_{k +3}+\left (3 a a_{k +1}+6 a_{k +3}\right ) k^{2}+\left (3 a^{2} a_{k -1}+3 a a_{k +1}+11 a_{k +3}\right ) k +a^{3} a_{k -3}-3 a^{2} 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 (3 a a_{k +4}+6 a_{k +6}\right ) \left (k +3\right )^{2}+\left (3 a^{2} a_{k +2}+3 a a_{k +4}+11 a_{k +6}\right ) \left (k +3\right )+a^{3} a_{k}-3 a^{2} 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 {a \left (a^{2} a_{k}+3 a k a_{k +2}+3 k^{2} a_{k +4}+6 a a_{k +2}+21 k a_{k +4}+36 a_{k +4}\right )}{k^{3}+15 k^{2}+74 k +120}, a_{3}=0, a_{4}=-\frac {a_{2} a}{4}, a_{5}=-\frac {a_{1} a^{2}}{20}\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 <- successful conversion to a linear ODE with constant coefficients`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 46
dsolve(diff(diff(diff(y(x),x),x),x)+3*a*x*diff(diff(y(x),x),x)+3*a^2*x^2*diff(y(x),x)+a^3*x^3*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {x \left (2 \sqrt {3}\, \sqrt {a}+a x \right )}{2}} \left (c_{2} {\mathrm e}^{2 \sqrt {3}\, \sqrt {a}\, x}+c_{1} {\mathrm e}^{\sqrt {3}\, \sqrt {a}\, x}+c_{3} \right ) \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 68
DSolve[a^3*x^3*y[x] + 3*a^2*x^2*y'[x] + 3*a*x*y''[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {a x^2}{2}-\sqrt {3} \sqrt {a} x} \left (c_1 e^{\sqrt {3} \sqrt {a} x}+c_3 e^{2 \sqrt {3} \sqrt {a} x}+c_2\right ) \]