Internal problem ID [6904]
Internal file name [OUTPUT/6147_Tuesday_August_09_2022_05_23_21_AM_91064150/index.tex]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th
edition. 1997.
Section: CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises
page 355
Problem number: 16.
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, _exact, _linear, _homogeneous]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime \prime }+x^{2} y^{\prime \prime }+5 x y^{\prime }+3 y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }+\left (\frac {d}{d x}y^{\prime }\right ) x^{2}+5 x y^{\prime }+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 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} 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} \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_{k} \left (k +3\right ) \left (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\right ) \left (k +1\right ) \left (k a_{k +3}+a_{k}+2 a_{k +3}\right )=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 {a_{k}}{k +2}\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 Louvillian solutions for 3rd order ODEs, imprimitive case: input is reducible, switching to DFactorsols checking if the LODE is of Euler type expon. solutions partially successful. Result(s) =`, [exp(-(1/3)*x^3)*x]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 98
dsolve(diff(y(x),x$3)+x^2*diff(y(x),x$2)+5*x*diff(y(x),x)+3*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (-c_{1} {\mathrm e}^{-\frac {x^{3}}{3}} x +c_{3} 3^{\frac {1}{3}}\right ) \left (-x^{3}\right )^{\frac {2}{3}}+x^{2} {\mathrm e}^{-\frac {x^{3}}{3}} \left (3 c_{2} \left (-x^{3}\right )^{\frac {1}{3}} \Gamma \left (\frac {1}{3}, -\frac {x^{3}}{3}\right ) \Gamma \left (\frac {2}{3}\right )-2 c_{2} \left (-x^{3}\right )^{\frac {1}{3}} \sqrt {3}\, \pi +x \Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right ) c_{3} -x c_{3} \Gamma \left (\frac {2}{3}\right )\right )}{\left (-x^{3}\right )^{\frac {2}{3}}} \]
✓ Solution by Mathematica
Time used: 0.093 (sec). Leaf size: 88
DSolve[y'''[x]+x^2*y''[x]+5*x*y'[x]+3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-\frac {x^3}{3}} \left (-2\ 3^{2/3} c_3 \sqrt [3]{-x^3} x \Gamma \left (-\frac {1}{3},-\frac {x^3}{3}\right )+3 \sqrt [3]{3} c_1 \left (-x^3\right )^{2/3} \Gamma \left (\frac {1}{3},-\frac {x^3}{3}\right )+18 c_2 x^2\right )}{18 x} \]