Internal problem ID [9780]
Internal file name [OUTPUT/8723_Monday_June_06_2022_05_20_21_AM_65393728/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1454.
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.
\[ \boxed {y^{\prime \prime \prime }+2 a x y^{\prime }+a y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }+2 a x y^{\prime }+a y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime } \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \\ {} & {} & y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k} \\ \square & {} & \textrm {Rewrite DE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x \cdot y^{\prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x \cdot y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} k \,x^{k} \\ {} & \circ & \textrm {Convert}\hspace {3pt} \frac {d}{d x}y^{\prime \prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }=\moverset {\infty }{\munderset {k =3}{\sum }}a_{k} k \left (k -1\right ) \left (k -2\right ) x^{k -3} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k +3} \left (k +3\right ) \left (k +2\right ) \left (k +1\right ) x^{k} \\ & {} & \textrm {Rewrite DE with series expansions}\hspace {3pt} \\ {} & {} & \moverset {\infty }{\munderset {k =0}{\sum }}\left (a_{k +3} \left (k +3\right ) \left (k +2\right ) \left (k +1\right )+a a_{k} \left (2 k +1\right )\right ) x^{k}=0 \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (k^{3}+6 k^{2}+11 k +6\right ) a_{k +3}+2 \left (k +\frac {1}{2}\right ) a a_{k}=0 \\ \bullet & {} & \textrm {Recursion relation that defines the series solution to the ODE}\hspace {3pt} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k}, a_{k +3}=-\frac {\left (2 k +1\right ) a a_{k}}{k^{3}+6 k^{2}+11 k +6}\right ] \end {array} \]
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 Equation is the 2nd symmetric power of diff(diff(y(x),x),x)+1/2*a*x*y(x) = 0 -> Attempting now to solve this lower order ODE trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 55
dsolve(diff(diff(diff(y(x),x),x),x)+2*a*x*diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {AiryAi}\left (-\frac {2^{\frac {2}{3}} a^{\frac {1}{3}} x}{2}\right )^{2}+c_{2} \operatorname {AiryBi}\left (-\frac {2^{\frac {2}{3}} a^{\frac {1}{3}} x}{2}\right )^{2}+c_{3} \operatorname {AiryAi}\left (-\frac {2^{\frac {2}{3}} a^{\frac {1}{3}} x}{2}\right ) \operatorname {AiryBi}\left (-\frac {2^{\frac {2}{3}} a^{\frac {1}{3}} x}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 79
DSolve[a*y[x] + 2*a*x*y'[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \operatorname {AiryAi}\left (\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a} x\right )^2+c_3 \operatorname {AiryBi}\left (\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a} x\right )^2+c_2 \operatorname {AiryAi}\left (\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a} x\right ) \operatorname {AiryBi}\left (\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a} x\right ) \]