Internal problem ID [9654]
Internal file name [OUTPUT/8596_Monday_June_06_2022_04_19_02_AM_84202337/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1327.
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.
\[ \boxed {y^{\prime \prime }-\frac {2 y^{\prime }}{x \left (x -2\right )}+\frac {y}{x^{2} \left (x -2\right )}=0} \]
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime }\right ) x^{2} \left (x -2\right )-2 x y^{\prime }+y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=\frac {2 y^{\prime }}{x \left (x -2\right )}-\frac {y}{x^{2} \left (x -2\right )} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }-\frac {2 y^{\prime }}{x \left (x -2\right )}+\frac {y}{x^{2} \left (x -2\right )}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=-\frac {2}{x \left (x -2\right )}, P_{3}\left (x \right )=\frac {1}{x^{2} \left (x -2\right )}\right ] \\ {} & \circ & x \cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x \cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=1 \\ {} & \circ & x^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=-\frac {1}{2} \\ {} & \circ & x =0\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime }\right ) x^{2} \left (x -2\right )-2 x y^{\prime }+y=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \\ {} & {} & y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r} \\ \square & {} & \textrm {Rewrite ODE 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} \left (k +r \right ) x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =2..3 \\ {} & {} & x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) x^{k +r -2+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2-m \\ {} & {} & x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =-2+m}{\sum }}a_{k +2-m} \left (k +2-m +r \right ) \left (k +1-m +r \right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & -a_{0} \left (2 r^{2}-1\right ) x^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (-a_{k} \left (2 k^{2}+4 k r +2 r^{2}-1\right )+a_{k -1} \left (k +r -1\right ) \left (k -2+r \right )\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & -2 r^{2}+1=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{-\frac {\sqrt {2}}{2}, \frac {\sqrt {2}}{2}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & a_{k -1} \left (k +r -1\right ) \left (k -2+r \right )-2 \left (k^{2}+2 k r +r^{2}-\frac {1}{2}\right ) a_{k}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & a_{k} \left (k +r \right ) \left (k +r -1\right )-2 \left (\left (k +1\right )^{2}+2 \left (k +1\right ) r +r^{2}-\frac {1}{2}\right ) a_{k +1}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=\frac {a_{k} \left (k +r \right ) \left (k +r -1\right )}{2 k^{2}+4 k r +2 r^{2}+4 k +4 r +1} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {\sqrt {2}}{2} \\ {} & {} & a_{k +1}=\frac {a_{k} \left (k -\frac {\sqrt {2}}{2}\right ) \left (k -\frac {\sqrt {2}}{2}-1\right )}{2 k^{2}-2 k \sqrt {2}+2+4 k -2 \sqrt {2}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\frac {\sqrt {2}}{2} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -\frac {\sqrt {2}}{2}}, a_{k +1}=\frac {a_{k} \left (k -\frac {\sqrt {2}}{2}\right ) \left (k -\frac {\sqrt {2}}{2}-1\right )}{2 k^{2}-2 k \sqrt {2}+2+4 k -2 \sqrt {2}}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {\sqrt {2}}{2} \\ {} & {} & a_{k +1}=\frac {a_{k} \left (k +\frac {\sqrt {2}}{2}\right ) \left (k +\frac {\sqrt {2}}{2}-1\right )}{2 k^{2}+2 k \sqrt {2}+2+4 k +2 \sqrt {2}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {\sqrt {2}}{2} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {\sqrt {2}}{2}}, a_{k +1}=\frac {a_{k} \left (k +\frac {\sqrt {2}}{2}\right ) \left (k +\frac {\sqrt {2}}{2}-1\right )}{2 k^{2}+2 k \sqrt {2}+2+4 k +2 \sqrt {2}}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -\frac {\sqrt {2}}{2}}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} x^{k +\frac {\sqrt {2}}{2}}\right ), a_{k +1}=\frac {a_{k} \left (k -\frac {\sqrt {2}}{2}\right ) \left (k -\frac {\sqrt {2}}{2}-1\right )}{2 k^{2}-2 k \sqrt {2}+2+4 k -2 \sqrt {2}}, b_{k +1}=\frac {b_{k} \left (k +\frac {\sqrt {2}}{2}\right ) \left (k +\frac {\sqrt {2}}{2}-1\right )}{2 k^{2}+2 k \sqrt {2}+2+4 k +2 \sqrt {2}}\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 <- hyper3 successful: received ODE is equivalent to the 2F1 ODE <- hypergeometric successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.234 (sec). Leaf size: 122
dsolve(diff(diff(y(x),x),x) = 2/x/(x-2)*diff(y(x),x)-1/x^2/(x-2)*y(x),y(x), singsol=all)
\[ y \left (x \right ) = 4 \left (x^{-\frac {\sqrt {2}}{2}}-x^{1-\frac {\sqrt {2}}{2}}+\frac {x^{2-\frac {\sqrt {2}}{2}}}{4}\right ) c_{1} \operatorname {hypergeom}\left (\left [2-\frac {\sqrt {2}}{2}, 1-\frac {\sqrt {2}}{2}\right ], \left [1-\sqrt {2}\right ], \frac {x}{2}\right )+\operatorname {hypergeom}\left (\left [2+\frac {\sqrt {2}}{2}, 1+\frac {\sqrt {2}}{2}\right ], \left [1+\sqrt {2}\right ], \frac {x}{2}\right ) c_{2} \left (x^{2+\frac {\sqrt {2}}{2}}+4 x^{\frac {\sqrt {2}}{2}}-4 x^{1+\frac {\sqrt {2}}{2}}\right ) \]
✓ Solution by Mathematica
Time used: 0.226 (sec). Leaf size: 105
DSolve[y''[x] == -(y[x]/((-2 + x)*x^2)) + (2*y'[x])/((-2 + x)*x),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (-\frac {1}{2}\right )^{-\frac {1}{\sqrt {2}}} x^{-\frac {1}{\sqrt {2}}} \left (\left (-\frac {1}{2}\right )^{\sqrt {2}} c_2 x^{\sqrt {2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{\sqrt {2}},-1+\frac {1}{\sqrt {2}},1+\sqrt {2},\frac {x}{2}\right )+c_1 \operatorname {Hypergeometric2F1}\left (-\frac {1}{\sqrt {2}},-1-\frac {1}{\sqrt {2}},1-\sqrt {2},\frac {x}{2}\right )\right ) \]