Internal problem ID [9684]
Internal file name [OUTPUT/8626_Monday_June_06_2022_04_30_34_AM_27274678/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1357.
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.
\[ \boxed {y^{\prime \prime }+\frac {\left (a \,x^{2}+a -1\right ) y^{\prime }}{x \left (x^{2}+1\right )}+\frac {\left (b \,x^{2}+c \right ) y}{x^{2} \left (x^{2}+1\right )}=0} \]
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x^{4}+x^{2}\right ) \left (\frac {d}{d x}y^{\prime }\right )+\left (a \,x^{2}+a -1\right ) x y^{\prime }+\left (b \,x^{2}+c \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 {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=-\frac {\left (a \,x^{2}+a -1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (b \,x^{2}+c \right ) y}{x^{2} \left (x^{2}+1\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 {\left (a \,x^{2}+a -1\right ) y^{\prime }}{x \left (x^{2}+1\right )}+\frac {\left (b \,x^{2}+c \right ) y}{x^{2} \left (x^{2}+1\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 {a \,x^{2}+a -1}{x \left (x^{2}+1\right )}, P_{3}\left (x \right )=\frac {b \,x^{2}+c}{x^{2} \left (x^{2}+1\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}}}=a -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}}}=c \\ {} & \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}+1\right )+\left (a \,x^{2}+a -1\right ) x y^{\prime }+\left (b \,x^{2}+c \right ) 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^{m}\cdot y\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..2 \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..3 \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =-1+m}{\sum }}a_{k +1-m} \left (k +1-m +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..4 \\ {} & {} & 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 (a r +r^{2}+c -2 r \right ) x^{r}+a_{1} \left (a r +r^{2}+a +c -1\right ) x^{1+r}+\left (\moverset {\infty }{\munderset {k =2}{\sum }}\left (a_{k} \left (a k +a r +k^{2}+2 k r +r^{2}+c -2 k -2 r \right )+a_{k -2} \left (a \left (k -2\right )+a r +\left (k -2\right )^{2}+2 \left (k -2\right ) r +r^{2}+b -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} \\ {} & {} & a r +r^{2}+c -2 r =0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}, -\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right \} \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & a_{1} \left (a r +r^{2}+a +c -1\right )=0 \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & a_{1}=0 \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (k^{2}+\left (a +2 r -5\right ) k +r^{2}+\left (a -5\right ) r -2 a +b +6\right ) a_{k -2}+\left (k^{2}+\left (a +2 r -2\right ) k +r^{2}+\left (a -2\right ) r +c \right ) a_{k}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & \left (\left (k +2\right )^{2}+\left (a +2 r -5\right ) \left (k +2\right )+r^{2}+\left (a -5\right ) r -2 a +b +6\right ) a_{k}+\left (\left (k +2\right )^{2}+\left (a +2 r -2\right ) \left (k +2\right )+r^{2}+\left (a -2\right ) r +c \right ) a_{k +2}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=-\frac {\left (a k +a r +k^{2}+2 k r +r^{2}+b -k -r \right ) a_{k}}{a k +a r +k^{2}+2 k r +r^{2}+2 a +c +2 k +2 r} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2} \\ {} & {} & a_{k +2}=-\frac {\left (a k +a \left (-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+k^{2}+2 k \left (-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+{\left (-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )}^{2}+b -k +\frac {a}{2}-1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right ) a_{k}}{a k +a \left (-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+k^{2}+2 k \left (-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+{\left (-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )}^{2}+a +c +2 k +2-\sqrt {a^{2}-4 a -4 c +4}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}}, a_{k +2}=-\frac {\left (a k +a \left (-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+k^{2}+2 k \left (-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+{\left (-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )}^{2}+b -k +\frac {a}{2}-1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right ) a_{k}}{a k +a \left (-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+k^{2}+2 k \left (-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+{\left (-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )}^{2}+a +c +2 k +2-\sqrt {a^{2}-4 a -4 c +4}}, a_{1}=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2} \\ {} & {} & a_{k +2}=-\frac {\left (a k +a \left (-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+k^{2}+2 k \left (-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+{\left (-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )}^{2}+b -k +\frac {a}{2}-1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right ) a_{k}}{a k +a \left (-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+k^{2}+2 k \left (-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+{\left (-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )}^{2}+a +c +2 k +2+\sqrt {a^{2}-4 a -4 c +4}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}}, a_{k +2}=-\frac {\left (a k +a \left (-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+k^{2}+2 k \left (-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+{\left (-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )}^{2}+b -k +\frac {a}{2}-1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right ) a_{k}}{a k +a \left (-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+k^{2}+2 k \left (-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+{\left (-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )}^{2}+a +c +2 k +2+\sqrt {a^{2}-4 a -4 c +4}}, a_{1}=0\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}d_{k} x^{k -\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}e_{k} x^{k -\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}}\right ), d_{k +2}=-\frac {\left (a k +a \left (-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+k^{2}+2 k \left (-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+{\left (-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )}^{2}+b -k +\frac {a}{2}-1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right ) d_{k}}{a k +a \left (-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+k^{2}+2 k \left (-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+{\left (-\frac {a}{2}+1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )}^{2}+a +c +2 k +2-\sqrt {a^{2}-4 a -4 c +4}}, d_{1}=0, e_{k +2}=-\frac {\left (a k +a \left (-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+k^{2}+2 k \left (-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+{\left (-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )}^{2}+b -k +\frac {a}{2}-1-\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right ) e_{k}}{a k +a \left (-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+k^{2}+2 k \left (-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )+{\left (-\frac {a}{2}+1+\frac {\sqrt {a^{2}-4 a -4 c +4}}{2}\right )}^{2}+a +c +2 k +2+\sqrt {a^{2}-4 a -4 c +4}}, e_{1}=0\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 <- heuristic approach successful <- hypergeometric successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.187 (sec). Leaf size: 97
dsolve(diff(diff(y(x),x),x) = -1/x*(a*x^2+a-1)/(x^2+1)*diff(y(x),x)-(b*x^2+c)/x^2/(x^2+1)*y(x),y(x), singsol=all)
\[ y \left (x \right ) = x^{1-\frac {a}{2}} \left (\operatorname {LegendreP}\left (-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}, \frac {\sqrt {a^{2}-4 a -4 c +4}}{2}, \sqrt {x^{2}+1}\right ) c_{1} +\operatorname {LegendreQ}\left (-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}, \frac {\sqrt {a^{2}-4 a -4 c +4}}{2}, \sqrt {x^{2}+1}\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.722 (sec). Leaf size: 264
DSolve[y''[x] == -(((c + b*x^2)*y[x])/(x^2*(1 + x^2))) - ((-1 + a + a*x^2)*y'[x])/(x*(1 + x^2)),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x^{-\frac {1}{2} \sqrt {a^2-4 a-4 c+4}-\frac {a}{2}+1} \left (c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (-\sqrt {a^2-2 a-4 b+1}-\sqrt {a^2-4 a-4 c+4}+1\right ),\frac {1}{4} \left (\sqrt {a^2-2 a-4 b+1}-\sqrt {a^2-4 a-4 c+4}+1\right ),1-\frac {1}{2} \sqrt {a^2-4 a-4 c+4},-x^2\right )+c_2 x^{\sqrt {a^2-4 a-4 c+4}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (-\sqrt {a^2-2 a-4 b+1}+\sqrt {a^2-4 a-4 c+4}+1\right ),\frac {1}{4} \left (\sqrt {a^2-2 a-4 b+1}+\sqrt {a^2-4 a-4 c+4}+1\right ),\frac {1}{2} \left (\sqrt {a^2-4 a-4 c+4}+2\right ),-x^2\right )\right ) \]