problem 3.48 (c)

Internal problem ID [5500]
Internal ﬁle name [OUTPUT/4748_Sunday_June_05_2022_03_04_54_PM_50846618/index.tex]

Book: Advanced Mathematical Methods for Scientists and Engineers, Bender and Orszag. Springer October 29, 1999
Section: Chapter 3. APPROXIMATE SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS. page 136
Problem number: 3.48 (c).
ODE order: 2.
ODE degree: 1.

The type(s) of ODE detected by this program : "second_order_bessel_ode", "second order series method. Irregular singular point"

\[ \boxed {x^{3} y^{\prime \prime }+y=\frac {1}{x^{4}}} \] With the expansion point for the power series method at \(x = 0\).

The type of the expansion point is ﬁrst determined. This is done on the homogeneous part of the ODE. \[ x^{3} y^{\prime \prime }+y = 0 \] The following is summary of singularities for the above ode. Writing the ode as \begin {align*} y^{\prime \prime }+p(x) y^{\prime } + q(x) y &=0 \end {align*}

Combining everything together gives the following summary of singularities for the ode as

Since \(x = 0\) is not an ordinary point, then we will now check if it is a regular singular point. Unable to solve since \(x = 0\) is not regular singular point. Terminating.

Veriﬁcation of solutions N/A

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 2; linear nonhomogeneous with symmetry [0,1] 
trying a double symmetry of the form [xi=0, eta=F(x)] 
-> Try solving first the homogeneous part of the ODE 
   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 
      <- Bessel successful 
   <- special function solution successful 
<- solving first the homogeneous part of the ODE successful`

\[ y(x)\to e^{-\frac {2 i}{\sqrt {x}}} x^{3/4} \left (\frac {33424574007825 x^5}{281474976710656}-\frac {468131288625 i x^{9/2}}{8796093022208}-\frac {14783093325 x^4}{549755813888}+\frac {66891825 i x^{7/2}}{4294967296}+\frac {2837835 x^3}{268435456}-\frac {72765 i x^{5/2}}{8388608}-\frac {4725 x^2}{524288}+\frac {105 i x^{3/2}}{8192}+\frac {15 x}{512}-\frac {3 i \sqrt {x}}{16}+1\right ) c_1+e^{\frac {2 i}{\sqrt {x}}} x^{3/4} \left (\frac {33424574007825 x^5}{281474976710656}+\frac {468131288625 i x^{9/2}}{8796093022208}-\frac {14783093325 x^4}{549755813888}-\frac {66891825 i x^{7/2}}{4294967296}+\frac {2837835 x^3}{268435456}+\frac {72765 i x^{5/2}}{8388608}-\frac {4725 x^2}{524288}-\frac {105 i x^{3/2}}{8192}+\frac {15 x}{512}+\frac {3 i \sqrt {x}}{16}+1\right ) c_2+\frac {\left (\frac {33424574007825 x^5}{281474976710656}+\frac {468131288625 i x^{9/2}}{8796093022208}-\frac {14783093325 x^4}{549755813888}-\frac {66891825 i x^{7/2}}{4294967296}+\frac {2837835 x^3}{268435456}+\frac {72765 i x^{5/2}}{8388608}-\frac {4725 x^2}{524288}-\frac {105 i x^{3/2}}{8192}+\frac {15 x}{512}+\frac {3 i \sqrt {x}}{16}+1\right ) \left (-45110302419831396543150980625 i x^{21/2}-42687836833427482392928732500 x^{10}-51974136213779750627466810000 i x^{19/2}+787128410789845519875480000 x^9+23504262853301237929117996800 i x^{17/2}-2844571059555743253185049600 x^8-16882400309820166719959961600 i x^{15/2}+14244707939052130467069542400 x^7+26274579672761392011514675200 i x^{13/2}-287333474777679866805355806720 x^6-2357805487104328892389014896640 i x^{11/2}-9431221948417315569556059586560 x^5+(13135986528809356661664114058199040+13135986528809356661664114058199040 i) e^{\frac {2 i}{\sqrt {x}}} \sqrt {\pi } \text {erf}\left (\frac {1+i}{\sqrt [4]{x}}\right ) x^{19/4}-52591304980570082036042064671539200 i x^{9/2}+70120300724414415842325758056857600 x^4+56096676899434227136229379617587200 i x^{7/2}-32055063435269639516813675986944000 x^3-14246790158231042731950078280335360 i x^{5/2}+5180586990275096078169557560197120 x^2+1594083009187391326184853272002560 i x^{3/2}-425019477807771039020566521577472 x-100144397418030122718239553224704 i \sqrt {x}+20282409603651670423947251286016\right )}{40564819207303340847894502572032 x^4}+\frac {e^{-\frac {2 i}{\sqrt {x}}} x^{3/4} \left (\frac {33424574007825 x^5}{281474976710656}-\frac {468131288625 i x^{9/2}}{8796093022208}-\frac {14783093325 x^4}{549755813888}+\frac {66891825 i x^{7/2}}{4294967296}+\frac {2837835 x^3}{268435456}-\frac {72765 i x^{5/2}}{8388608}-\frac {4725 x^2}{524288}+\frac {105 i x^{3/2}}{8192}+\frac {15 x}{512}-\frac {3 i \sqrt {x}}{16}+1\right ) \left (\frac {4 e^{\frac {2 i}{\sqrt {x}}} \left (45110302419831396543150980625 i x^{21/2}-42687836833427482392928732500 x^{10}+51974136213779750627466810000 i x^{19/2}+787128410789845519875480000 x^9-23504262853301237929117996800 i x^{17/2}-2844571059555743253185049600 x^8+16882400309820166719959961600 i x^{15/2}+14244707939052130467069542400 x^7-26274579672761392011514675200 i x^{13/2}-287333474777679866805355806720 x^6+2357805487104328892389014896640 i x^{11/2}-9431221948417315569556059586560 x^5+52591304980570082036042064671539200 i x^{9/2}+70120300724414415842325758056857600 x^4-56096676899434227136229379617587200 i x^{7/2}-32055063435269639516813675986944000 x^3+14246790158231042731950078280335360 i x^{5/2}+5180586990275096078169557560197120 x^2-1594083009187391326184853272002560 i x^{3/2}-425019477807771039020566521577472 x+100144397418030122718239553224704 i \sqrt {x}+20282409603651670423947251286016\right )}{x^{19/4}}-(52543946115237426646656456232796160+52543946115237426646656456232796160 i) \sqrt {\pi } \text {erfi}\left (\frac {1+i}{\sqrt [4]{x}}\right )\right )}{162259276829213363391578010288128} \]


\(q(x)=\frac {1}{x^{3}}\)

singularity	type

\(x = 0\)	\(\text {``irregular''}\)

1.21 problem 3.48 (c)