2.41 problem Problem 18(f)

Internal problem ID [12261]
Internal file name [OUTPUT/10914_Thursday_September_28_2023_01_08_38_AM_56532464/index.tex]

Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section: Chapter 4, Second and Higher Order Linear Differential Equations. Problems page 221
Problem number: Problem 18(f).
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.

\[ \boxed {x^{2} y^{\prime \prime }+x^{2} y^{\prime }+2 \left (1-x \right ) y=0} \]

2.41.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d}{d x}y^{\prime }\right )+x^{2} y^{\prime }+\left (-2 x +2\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 }=-y^{\prime }+\frac {2 \left (x -1\right ) y}{x^{2}} \\ \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 }+y^{\prime }-\frac {2 \left (x -1\right ) 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 )=1, P_{3}\left (x \right )=-\frac {2 \left (x -1\right )}{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}}}=0 \\ {} & \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}}}=2 \\ {} & \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 )+x^{2} y^{\prime }+\left (-2 x +2\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..1 \\ {} & {} & 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^{2}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{2}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r +1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -1 \\ {} & {} & x^{2}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =1}{\sum }}a_{k -1} \left (k -1+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 -1+r \right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} \left (r^{2}-r +2\right ) x^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (a_{k} \left (k^{2}+2 k r +r^{2}-k -r +2\right )+a_{k -1} \left (k -3+r \right )\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & r^{2}-r +2=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}, \frac {1}{2}+\frac {\mathrm {I} \sqrt {7}}{2}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (k^{2}+\left (2 r -1\right ) k +r^{2}-r +2\right ) a_{k}+a_{k -1} \left (k -3+r \right )=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & \left (\left (k +1\right )^{2}+\left (2 r -1\right ) \left (k +1\right )+r^{2}-r +2\right ) a_{k +1}+a_{k} \left (k +r -2\right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=-\frac {a_{k} \left (k +r -2\right )}{k^{2}+2 k r +r^{2}+k +r +2} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2} \\ {} & {} & a_{k +1}=-\frac {a_{k} \left (k -\frac {3}{2}-\frac {\mathrm {I} \sqrt {7}}{2}\right )}{k^{2}+2 k \left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}\right )+\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}\right )^{2}+k +\frac {5}{2}-\frac {\mathrm {I} \sqrt {7}}{2}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}}, a_{k +1}=-\frac {a_{k} \left (k -\frac {3}{2}-\frac {\mathrm {I} \sqrt {7}}{2}\right )}{k^{2}+2 k \left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}\right )+\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}\right )^{2}+k +\frac {5}{2}-\frac {\mathrm {I} \sqrt {7}}{2}}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {1}{2}+\frac {\mathrm {I} \sqrt {7}}{2} \\ {} & {} & a_{k +1}=-\frac {a_{k} \left (k -\frac {3}{2}+\frac {\mathrm {I} \sqrt {7}}{2}\right )}{k^{2}+2 k \left (\frac {1}{2}+\frac {\mathrm {I} \sqrt {7}}{2}\right )+\left (\frac {1}{2}+\frac {\mathrm {I} \sqrt {7}}{2}\right )^{2}+k +\frac {5}{2}+\frac {\mathrm {I} \sqrt {7}}{2}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {1}{2}+\frac {\mathrm {I} \sqrt {7}}{2} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {1}{2}+\frac {\mathrm {I} \sqrt {7}}{2}}, a_{k +1}=-\frac {a_{k} \left (k -\frac {3}{2}+\frac {\mathrm {I} \sqrt {7}}{2}\right )}{k^{2}+2 k \left (\frac {1}{2}+\frac {\mathrm {I} \sqrt {7}}{2}\right )+\left (\frac {1}{2}+\frac {\mathrm {I} \sqrt {7}}{2}\right )^{2}+k +\frac {5}{2}+\frac {\mathrm {I} \sqrt {7}}{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 {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} x^{k +\frac {1}{2}+\frac {\mathrm {I} \sqrt {7}}{2}}\right ), a_{k +1}=-\frac {a_{k} \left (k -\frac {3}{2}-\frac {\mathrm {I} \sqrt {7}}{2}\right )}{k^{2}+2 k \left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}\right )+\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {7}}{2}\right )^{2}+k +\frac {5}{2}-\frac {\mathrm {I} \sqrt {7}}{2}}, b_{k +1}=-\frac {b_{k} \left (k -\frac {3}{2}+\frac {\mathrm {I} \sqrt {7}}{2}\right )}{k^{2}+2 k \left (\frac {1}{2}+\frac {\mathrm {I} \sqrt {7}}{2}\right )+\left (\frac {1}{2}+\frac {\mathrm {I} \sqrt {7}}{2}\right )^{2}+k +\frac {5}{2}+\frac {\mathrm {I} \sqrt {7}}{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: 0.063 (sec). Leaf size: 93

dsolve(x^2*diff(y(x),x$2)+x^2*diff(y(x),x)+2*(1-x)*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = \sqrt {x}\, {\mathrm e}^{-\frac {x}{2}} \left (x c_{1} \left (x +2\right ) \operatorname {BesselI}\left (\frac {i \sqrt {7}}{2}+1, \frac {x}{2}\right )-x c_{2} \left (x +2\right ) \operatorname {BesselK}\left (\frac {i \sqrt {7}}{2}+1, \frac {x}{2}\right )+\left (\operatorname {BesselI}\left (\frac {i \sqrt {7}}{2}, \frac {x}{2}\right ) c_{1} +\operatorname {BesselK}\left (\frac {i \sqrt {7}}{2}, \frac {x}{2}\right ) c_{2} \right ) \left (-2+i \left (x +2\right ) \sqrt {7}+x^{2}+3 x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.085 (sec). Leaf size: 89

DSolve[x^2*y''[x]+x^2*y'[x]+2*(1-x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-x} x^{\frac {1}{2}+\frac {i \sqrt {7}}{2}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {5}{2}+\frac {i \sqrt {7}}{2},1+i \sqrt {7},x\right )+c_2 L_{-\frac {1}{2} i \left (-5 i+\sqrt {7}\right )}^{i \sqrt {7}}(x)\right ) \]