Internal problem ID [9524]
Internal file name [OUTPUT/8464_Monday_June_06_2022_03_08_28_AM_69611509/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1195.
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.
\[ \boxed {x^{2} y^{\prime \prime }+\left (x +3\right ) x y^{\prime }-y=0} \]
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d}{d x}y^{\prime }\right )+\left (x^{2}+3 x \right ) 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 {y}{x^{2}}-\frac {\left (x +3\right ) y^{\prime }}{x} \\ \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 (x +3\right ) y^{\prime }}{x}-\frac {y}{x^{2}}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=\frac {x +3}{x}, P_{3}\left (x \right )=-\frac {1}{x^{2}}\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}}}=3 \\ {} & \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}}}=-1 \\ {} & \circ & x =0\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d}{d x}y^{\prime }\right )+\left (x +3\right ) 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^{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 =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^{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} \left (k +r \right ) \left (k +r -1\right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} \left (r^{2}+2 r -1\right ) x^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (a_{k} \left (k^{2}+2 k r +r^{2}+2 k +2 r -1\right )+a_{k -1} \left (k +r -1\right )\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & r^{2}+2 r -1=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{-1-\sqrt {2}, \sqrt {2}-1\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (k^{2}+\left (2 r +2\right ) k +r^{2}+2 r -1\right ) a_{k}+a_{k -1} \left (k +r -1\right )=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & \left (\left (k +1\right )^{2}+\left (2 r +2\right ) \left (k +1\right )+r^{2}+2 r -1\right ) a_{k +1}+a_{k} \left (k +r \right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=-\frac {a_{k} \left (k +r \right )}{k^{2}+2 k r +r^{2}+4 k +4 r +2} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-1-\sqrt {2} \\ {} & {} & a_{k +1}=-\frac {a_{k} \left (k -1-\sqrt {2}\right )}{k^{2}+2 k \left (-1-\sqrt {2}\right )+\left (-1-\sqrt {2}\right )^{2}+4 k -2-4 \sqrt {2}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-1-\sqrt {2} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -1-\sqrt {2}}, a_{k +1}=-\frac {a_{k} \left (k -1-\sqrt {2}\right )}{k^{2}+2 k \left (-1-\sqrt {2}\right )+\left (-1-\sqrt {2}\right )^{2}+4 k -2-4 \sqrt {2}}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\sqrt {2}-1 \\ {} & {} & a_{k +1}=-\frac {a_{k} \left (k +\sqrt {2}-1\right )}{k^{2}+2 k \left (\sqrt {2}-1\right )+\left (\sqrt {2}-1\right )^{2}+4 k +4 \sqrt {2}-2} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\sqrt {2}-1 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\sqrt {2}-1}, a_{k +1}=-\frac {a_{k} \left (k +\sqrt {2}-1\right )}{k^{2}+2 k \left (\sqrt {2}-1\right )+\left (\sqrt {2}-1\right )^{2}+4 k +4 \sqrt {2}-2}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -1-\sqrt {2}}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} x^{k +\sqrt {2}-1}\right ), a_{k +1}=-\frac {a_{k} \left (k -1-\sqrt {2}\right )}{k^{2}+2 k \left (-1-\sqrt {2}\right )+\left (-1-\sqrt {2}\right )^{2}+4 k -2-4 \sqrt {2}}, b_{k +1}=-\frac {b_{k} \left (k +\sqrt {2}-1\right )}{k^{2}+2 k \left (\sqrt {2}-1\right )+\left (\sqrt {2}-1\right )^{2}+4 k +4 \sqrt {2}-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 successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 93
dsolve(x^2*diff(diff(y(x),x),x)+(x+3)*x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (-c_{1} \left (\sqrt {2}+x +1\right ) \operatorname {BesselI}\left (-\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right )-c_{1} \left (-\sqrt {2}+x +1\right ) \operatorname {BesselI}\left (\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right )+c_{2} \left (\left (-\sqrt {2}-x -1\right ) \operatorname {BesselK}\left (-\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right )+\operatorname {BesselK}\left (\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right ) \left (-\sqrt {2}+x +1\right )\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.049 (sec). Leaf size: 63
DSolve[-y[x] + x*(3 + x)*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-x} x^{\sqrt {2}-1} \left (c_1 \operatorname {HypergeometricU}\left (2+\sqrt {2},1+2 \sqrt {2},x\right )+c_2 L_{-2-\sqrt {2}}^{2 \sqrt {2}}(x)\right ) \]