Internal problem ID [9869]
Internal file name [OUTPUT/8815_Monday_June_06_2022_05_32_50_AM_35619943/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1546.
ODE order: 4.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_high_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime \prime \prime }+4 a x y^{\prime \prime \prime }+6 a^{2} x^{2} y^{\prime \prime }+4 a^{3} x^{3} y^{\prime }+a^{4} x^{4} y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime }+4 a x \left (\frac {d}{d x}y^{\prime \prime }\right )+6 a^{2} x^{2} \left (\frac {d}{d x}y^{\prime }\right )+4 a^{3} x^{3} y^{\prime }+a^{4} x^{4} y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & \frac {d}{d x}\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^{4}\cdot y\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{4}\cdot y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +4} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -4 \\ {} & {} & x^{4}\cdot y=\moverset {\infty }{\munderset {k =4}{\sum }}a_{k -4} x^{k} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{3}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{3}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} k \,x^{k +2} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -2 \\ {} & {} & x^{3}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =2}{\sum }}a_{k -2} \left (k -2\right ) x^{k} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{2}\cdot \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{2}\cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} k \left (k -1\right ) x^{k} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x \cdot \left (\frac {d}{d x}y^{\prime \prime }\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x \cdot \left (\frac {d}{d x}y^{\prime \prime }\right )=\moverset {\infty }{\munderset {k =2}{\sum }}a_{k} k \left (k -1\right ) \left (k -2\right ) x^{k -2} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & x \cdot \left (\frac {d}{d x}y^{\prime \prime }\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k +2} \left (k +2\right ) \left (k +1\right ) k \,x^{k} \\ {} & \circ & \textrm {Convert}\hspace {3pt} \frac {d}{d x}\frac {d}{d x}y^{\prime \prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime }=\moverset {\infty }{\munderset {k =4}{\sum }}a_{k} k \left (k -1\right ) \left (k -2\right ) \left (k -3\right ) x^{k -4} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +4 \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k +4} \left (k +4\right ) \left (k +3\right ) \left (k +2\right ) \left (k +1\right ) x^{k} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & 24 a_{4}+\left (24 a_{3} a +120 a_{5}\right ) x +\left (12 a_{2} a^{2}+96 a a_{4}+360 a_{6}\right ) x^{2}+\left (4 a_{1} a^{3}+36 a_{3} a^{2}+240 a a_{5}+840 a_{7}\right ) x^{3}+\left (\moverset {\infty }{\munderset {k =4}{\sum }}\left (a_{k +4} \left (k +4\right ) \left (k +3\right ) \left (k +2\right ) \left (k +1\right )+4 a a_{k +2} \left (k +2\right ) \left (k +1\right ) k +6 a^{2} a_{k} k \left (k -1\right )+4 a^{3} a_{k -2} \left (k -2\right )+a^{4} a_{k -4}\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 [24 a_{4}=0, 24 a_{3} a +120 a_{5}=0, 12 a_{2} a^{2}+96 a a_{4}+360 a_{6}=0, 4 a_{1} a^{3}+36 a_{3} a^{2}+240 a a_{5}+840 a_{7}=0\right ] \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & \left \{a_{4}=0, a_{5}=-\frac {a_{3} a}{5}, a_{6}=-\frac {a_{2} a^{2}}{30}, a_{7}=-\frac {1}{210} a_{1} a^{3}+\frac {1}{70} a_{3} a^{2}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & k^{4} a_{k +4}+\left (4 a a_{k +2}+10 a_{k +4}\right ) k^{3}+\left (6 a^{2} a_{k}+12 a a_{k +2}+35 a_{k +4}\right ) k^{2}+\left (4 a^{3} a_{k -2}-6 a^{2} a_{k}+8 a a_{k +2}+50 a_{k +4}\right ) k +a^{4} a_{k -4}-8 a^{3} a_{k -2}+24 a_{k +4}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +4 \\ {} & {} & \left (k +4\right )^{4} a_{k +8}+\left (4 a a_{k +6}+10 a_{k +8}\right ) \left (k +4\right )^{3}+\left (6 a^{2} a_{k +4}+12 a a_{k +6}+35 a_{k +8}\right ) \left (k +4\right )^{2}+\left (4 a^{3} a_{k +2}-6 a^{2} a_{k +4}+8 a a_{k +6}+50 a_{k +8}\right ) \left (k +4\right )+a^{4} a_{k}-8 a^{3} a_{k +2}+24 a_{k +8}=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 +8}=-\frac {a \left (a^{3} a_{k}+4 a^{2} k a_{k +2}+6 a \,k^{2} a_{k +4}+4 k^{3} a_{k +6}+8 a^{2} a_{k +2}+42 a k a_{k +4}+60 k^{2} a_{k +6}+72 a a_{k +4}+296 k a_{k +6}+480 a_{k +6}\right )}{k^{4}+26 k^{3}+251 k^{2}+1066 k +1680}, a_{4}=0, a_{5}=-\frac {a_{3} a}{5}, a_{6}=-\frac {a_{2} a^{2}}{30}, a_{7}=-\frac {1}{210} a_{1} a^{3}+\frac {1}{70} a_{3} a^{2}\right ] \end {array} \]
Maple trace
`Methods for high 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: 126
dsolve(diff(diff(diff(diff(y(x),x),x),x),x)+4*a*x*diff(diff(diff(y(x),x),x),x)+6*a^2*x^2*diff(diff(y(x),x),x)+4*a^3*x^3*diff(y(x),x)+a^4*x^4*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {x \left (2 \sqrt {3+\sqrt {6}}\, \sqrt {a}+2 \sqrt {3-\sqrt {6}}\, \sqrt {a}+a x \right )}{2}} \left (c_{2} {\mathrm e}^{\sqrt {a}\, x \left (\sqrt {3+\sqrt {6}}+2 \sqrt {3-\sqrt {6}}\right )}+c_{4} {\mathrm e}^{\sqrt {a}\, x \left (2 \sqrt {3+\sqrt {6}}+\sqrt {3-\sqrt {6}}\right )}+c_{3} {\mathrm e}^{\sqrt {3-\sqrt {6}}\, \sqrt {a}\, x}+c_{1} {\mathrm e}^{\sqrt {3+\sqrt {6}}\, \sqrt {a}\, x}\right ) \]
✓ Solution by Mathematica
Time used: 0.737 (sec). Leaf size: 165
DSolve[a^4*x^4*y[x] + 4*a^3*x^3*y'[x] + 6*a^2*x^2*y''[x] + 4*a*x*Derivative[3][y][x] + Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-\frac {a x^2}{2}-\sqrt {3+\sqrt {6}} \sqrt {a} x} \left (6 a \left (c_1 e^{\frac {\left (-3+\sqrt {3}+\sqrt {6}\right ) a x}{\sqrt {-\left (\left (\sqrt {6}-3\right ) a\right )}}}+c_2 e^{\frac {\left (3+\sqrt {3}-\sqrt {6}\right ) a x}{\sqrt {-\left (\left (\sqrt {6}-3\right ) a\right )}}}\right )+\sqrt {6} \sqrt {-\left (\left (\sqrt {6}-3\right ) a\right )} \left (c_4 e^{\frac {2 a x}{\sqrt {a-\sqrt {\frac {2}{3}} a}}}+c_3\right )\right )}{6 a} \]