Internal problem ID [10860]
Internal file name [OUTPUT/10117_Sunday_December_24_2023_05_12_38_PM_85001203/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-2 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 37.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y=0} \]
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }+x \left (a x +b \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\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^{m}\cdot y\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..2 \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =\max \left (0, -m \right )}{\sum }}a_{k} x^{k +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =\max \left (0, -m \right )+m}{\sum }}a_{k -m} x^{k} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..2 \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =\max \left (0, 1-m \right )}{\sum }}a_{k} k \,x^{k -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =\max \left (0, 1-m \right )+m -1}{\sum }}a_{k +1-m} \left (k +1-m \right ) x^{k} \\ {} & \circ & \textrm {Convert}\hspace {3pt} \frac {d}{d x}y^{\prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=\moverset {\infty }{\munderset {k =2}{\sum }}a_{k} k \left (k -1\right ) x^{k -2} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & \frac {d}{d x}y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k +2} \left (k +2\right ) \left (k +1\right ) x^{k} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} \gamma +2 a_{2}+\left (6 a_{3}+a_{1} \left (b +\gamma \right )+a_{0} \beta \right ) x +\left (\moverset {\infty }{\munderset {k =2}{\sum }}\left (a_{k +2} \left (k +2\right ) \left (k +1\right )+a_{k} \left (b k +\gamma \right )+a_{k -1} \left (a \left (k -1\right )+\beta \right )+a_{k -2} \alpha \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 [2 a_{2}+a_{0} \gamma =0, 6 a_{3}+a_{1} \left (b +\gamma \right )+a_{0} \beta =0\right ] \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & \left \{a_{2}=-\frac {a_{0} \gamma }{2}, a_{3}=-\frac {1}{6} a_{1} b -\frac {1}{6} a_{0} \beta -\frac {1}{6} a_{1} \gamma \right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & k^{2} a_{k +2}+\left (a a_{k -1}+b a_{k}+3 a_{k +2}\right ) k +\left (-a +\beta \right ) a_{k -1}+a_{k -2} \alpha +a_{k} \gamma +2 a_{k +2}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & \left (k +2\right )^{2} a_{k +4}+\left (a a_{k +1}+b a_{k +2}+3 a_{k +4}\right ) \left (k +2\right )+\left (-a +\beta \right ) a_{k +1}+a_{k} \alpha +a_{k +2} \gamma +2 a_{k +4}=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 a_{k +1}+b k a_{k +2}+a a_{k +1}+a_{k} \alpha +2 b a_{k +2}+\beta a_{k +1}+a_{k +2} \gamma }{k^{2}+7 k +12}, a_{2}=-\frac {a_{0} \gamma }{2}, a_{3}=-\frac {1}{6} a_{1} b -\frac {1}{6} a_{0} \beta -\frac {1}{6} a_{1} \gamma \right ] \end {array} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- 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 -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius trying a solution in terms of MeijerG functions -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius <- Heun successful: received ODE is equivalent to the HeunT ODE, case c = 0 `
✓ Solution by Maple
Time used: 0.265 (sec). Leaf size: 271
dsolve(diff(y(x),x$2)+(a*x^2+b*x)*diff(y(x),x)+(alpha*x^2+beta*x+gamma)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {\operatorname {csgn}\left (a \right ) x \left (2 a^{2} x^{2} \operatorname {csgn}\left (a \right )+3 a b x \,\operatorname {csgn}\left (a \right )+2 a^{2} x^{2}+3 a b x -12 \alpha \right )}{12 a}} \operatorname {HeunT}\left (\frac {3^{\frac {2}{3}} \left (2 a^{2} \gamma -a b \beta +\alpha \,b^{2}+2 \alpha ^{2}\right )}{2 a^{2} \left (a^{2}\right )^{\frac {1}{3}}}, -\frac {3 \left (a^{2}-\beta a +b \alpha \right ) \operatorname {csgn}\left (a \right )}{a^{2}}, -\frac {3^{\frac {1}{3}} \left (b^{2}+8 \alpha \right )}{4 \left (a^{2}\right )^{\frac {2}{3}}}, \frac {3^{\frac {2}{3}} a \left (2 a x +b \right )}{6 \left (a^{2}\right )^{\frac {5}{6}}}\right )+c_{2} {\mathrm e}^{-\frac {\operatorname {csgn}\left (a \right ) x \left (2 a^{2} x^{2} \operatorname {csgn}\left (a \right )+3 a b x \,\operatorname {csgn}\left (a \right )-2 a^{2} x^{2}-3 a b x +12 \alpha \right )}{12 a}} \operatorname {HeunT}\left (\frac {3^{\frac {2}{3}} \left (2 a^{2} \gamma -a b \beta +\alpha \,b^{2}+2 \alpha ^{2}\right )}{2 a^{2} \left (a^{2}\right )^{\frac {1}{3}}}, \frac {3 \left (a^{2}-\beta a +b \alpha \right ) \operatorname {csgn}\left (a \right )}{a^{2}}, -\frac {3^{\frac {1}{3}} \left (b^{2}+8 \alpha \right )}{4 \left (a^{2}\right )^{\frac {2}{3}}}, -\frac {3^{\frac {2}{3}} a \left (2 a x +b \right )}{6 \left (a^{2}\right )^{\frac {5}{6}}}\right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y''[x]+(a*x^2+b*x)*y'[x]+(\[Alpha]*x^2+\[Beta]*x+\[Gamma])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved