50.22.20 problem 3(d)
Internal
problem
ID
[8163]
Book
:
Differential
Equations:
Theory,
Technique,
and
Practice
by
George
Simmons,
Steven
Krantz.
McGraw-Hill
NY.
2007.
1st
Edition.
Section
:
Chapter
4.
Power
Series
Solutions
and
Special
Functions.
Problems
for
review
and
discovert.
(A)
Drill
Exercises
.
Page
194
Problem
number
:
3(d)
Date
solved
:
Sunday, March 30, 2025 at 12:47:05 PM
CAS
classification
:
[[_3rd_order, _with_linear_symmetries]]
\begin{align*} x^{3} y^{\prime \prime \prime }+\left (2 x^{3}-x^{2}\right ) y^{\prime \prime }-x y^{\prime }+y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.031 (sec). Leaf size: 360
Order:=8;
ode:=x^3*diff(diff(diff(y(x),x),x),x)+(2*x^3-x^2)*diff(diff(y(x),x),x)-x*diff(y(x),x)+y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = c_3 x \left (1+\operatorname {O}\left (x^{8}\right )\right )+c_2 \,x^{\frac {3}{2}-\frac {\sqrt {13}}{2}} \left (1-x +\frac {-3+\sqrt {13}}{-4+2 \sqrt {13}} x^{2}+\frac {5-\sqrt {13}}{-12+6 \sqrt {13}} x^{3}+\frac {1}{24} \frac {\left (-5+\sqrt {13}\right ) \left (-7+\sqrt {13}\right )}{\left (-2+\sqrt {13}\right ) \left (-4+\sqrt {13}\right )} x^{4}+\frac {1}{30} \frac {-19+4 \sqrt {13}}{\left (-2+\sqrt {13}\right ) \left (-4+\sqrt {13}\right )} x^{5}+\frac {1}{20} \frac {-29+7 \sqrt {13}}{\left (-2+\sqrt {13}\right ) \left (-4+\sqrt {13}\right ) \left (-6+\sqrt {13}\right )} x^{6}+\frac {-\frac {117}{35}+\frac {6 \sqrt {13}}{7}}{\left (-2+\sqrt {13}\right ) \left (-4+\sqrt {13}\right ) \left (-6+\sqrt {13}\right ) \left (-7+\sqrt {13}\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_1 \,x^{\frac {3}{2}+\frac {\sqrt {13}}{2}} \left (1-x +\frac {3+\sqrt {13}}{4+2 \sqrt {13}} x^{2}+\frac {-5-\sqrt {13}}{12+6 \sqrt {13}} x^{3}+\frac {1}{24} \frac {\left (5+\sqrt {13}\right ) \left (7+\sqrt {13}\right )}{\left (2+\sqrt {13}\right ) \left (4+\sqrt {13}\right )} x^{4}-\frac {1}{30} \frac {19+4 \sqrt {13}}{\left (2+\sqrt {13}\right ) \left (4+\sqrt {13}\right )} x^{5}+\frac {1}{20} \frac {29+7 \sqrt {13}}{\left (2+\sqrt {13}\right ) \left (4+\sqrt {13}\right ) \left (6+\sqrt {13}\right )} x^{6}+\frac {-\frac {117}{35}-\frac {6 \sqrt {13}}{7}}{\left (2+\sqrt {13}\right ) \left (4+\sqrt {13}\right ) \left (6+\sqrt {13}\right ) \left (7+\sqrt {13}\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right )
\]
✓ Mathematica. Time used: 0.238 (sec). Leaf size: 310
ode=x^3*D[y[x],{x,3}]+(2*x^3-x^2)*D[y[x],{x,2}]-D[y[x],x]+y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to c_1 \left (\frac {99473 x^7}{1008}+\frac {1043 x^6}{144}+\frac {19 x^5}{24}+\frac {11 x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}+x+1\right )+c_2 e^{-\frac {2}{\sqrt {x}}} \left (-\frac {279112936065458899252220570230691 x^{13/2}}{160251477454333302276096}-\frac {2430057902534044595693470483 x^{11/2}}{100317681699677798400}-\frac {1545013796231079344731 x^{9/2}}{3562417673994240}-\frac {2005991558758787 x^{7/2}}{193273528320}-\frac {43999069453 x^{5/2}}{125829120}-\frac {438565 x^{3/2}}{24576}+\frac {14436319972596450047835320516938615783 x^7}{897408273744266492746137600}+\frac {3840864007433053956366665361751 x^6}{19260994886338137292800}+\frac {1786308115320202497636167 x^5}{569986827839078400}+\frac {319234145332261451 x^4}{4947802324992}+\frac {21959100963217 x^3}{12079595520}+\frac {117706529 x^2}{1572864}+\frac {2353 x}{512}-\frac {29 \sqrt {x}}{16}+1\right ) x^{11/4}+c_3 e^{\frac {2}{\sqrt {x}}} \left (\frac {279112936065458899252220570230691 x^{13/2}}{160251477454333302276096}+\frac {2430057902534044595693470483 x^{11/2}}{100317681699677798400}+\frac {1545013796231079344731 x^{9/2}}{3562417673994240}+\frac {2005991558758787 x^{7/2}}{193273528320}+\frac {43999069453 x^{5/2}}{125829120}+\frac {438565 x^{3/2}}{24576}+\frac {14436319972596450047835320516938615783 x^7}{897408273744266492746137600}+\frac {3840864007433053956366665361751 x^6}{19260994886338137292800}+\frac {1786308115320202497636167 x^5}{569986827839078400}+\frac {319234145332261451 x^4}{4947802324992}+\frac {21959100963217 x^3}{12079595520}+\frac {117706529 x^2}{1572864}+\frac {2353 x}{512}+\frac {29 \sqrt {x}}{16}+1\right ) x^{11/4}
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(1 - x)*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
Series solution not supported for ode of order > 2