42.1.21 problem 3.48 (c)
Internal
problem
ID
[6843]
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)
Date
solved
:
Monday, January 27, 2025 at 02:31:34 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} x^{3} y^{\prime \prime }+y&=\frac {1}{x^{4}} \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✗ Solution by Maple
Order:=6;
dsolve(x^3*diff(y(x),x$2)+y(x)=1/x^4,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.563 (sec). Leaf size: 873
AsymptoticDSolveValue[x^3*D[y[x],{x,2}]+y[x]==1/x^4,y[x],{x,0,"6"-1}]
\[
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}-183075518545920 \left (51515473086324893+\frac {71751737387620818787 \sqrt [4]{-1} e^{\frac {2 i}{\sqrt {x}}} \sqrt {2 \pi }}{\sqrt [4]{x}}\right ) x^5+13135986528809356661664114058199040 \sqrt [4]{-1} e^{\frac {2 i}{\sqrt {x}}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt [4]{-1} \sqrt {2}}{\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}}} \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 (52543946115237426646656456232796160 (-1)^{3/4} \sqrt {2 \pi } \text {erf}\left (\frac {(-1)^{3/4} \sqrt {2}}{\sqrt [4]{x}}\right ) x^{19/4}+52543946115237426646656456232796160 (-1)^{3/4} \sqrt {2 \pi } x^{19/4}+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 )\right )}{162259276829213363391578010288128 x^4}
\]