50.19.9 problem 2(e)
Internal
problem
ID
[8117]
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.
Section
4.4.
REGULAR
SINGULAR
POINTS.
Page
175
Problem
number
:
2(e)
Date
solved
:
Sunday, March 30, 2025 at 12:45:51 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{4} y^{\prime \prime }+\sin \left (x \right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✗ Maple
Order:=8;
ode:=x^4*diff(diff(y(x),x),x)+sin(x)*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Mathematica. Time used: 0.119 (sec). Leaf size: 294
ode=x^4*D[y[x],{x,2}]+Sin[x]*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to c_1 e^{-\frac {2 i}{\sqrt {x}}} x^{3/4} \left (\frac {16487484152477478659746223 i x^{13/2}}{2773583263632691770163200}-\frac {4594934148364735183693 i x^{11/2}}{6320013947079701299200}+\frac {12579783586699513 i x^{9/2}}{96185277197844480}-\frac {21896783401 i x^{7/2}}{579820584960}+\frac {856783 i x^{5/2}}{41943040}-\frac {3151 i x^{3/2}}{73728}-\frac {3986263268940827572255963529 x^7}{207094217017907652172185600}+\frac {21730712888356628741772337 x^6}{10920984100553723845017600}-\frac {1500040357444099007 x^5}{5129881450551705600}+\frac {4885269094757 x^4}{74217034874880}-\frac {2835642457 x^3}{108716359680}+\frac {11659 x^2}{524288}+\frac {15 x}{512}-\frac {3 i \sqrt {x}}{16}+1\right )+c_2 e^{\frac {2 i}{\sqrt {x}}} x^{3/4} \left (-\frac {16487484152477478659746223 i x^{13/2}}{2773583263632691770163200}+\frac {4594934148364735183693 i x^{11/2}}{6320013947079701299200}-\frac {12579783586699513 i x^{9/2}}{96185277197844480}+\frac {21896783401 i x^{7/2}}{579820584960}-\frac {856783 i x^{5/2}}{41943040}+\frac {3151 i x^{3/2}}{73728}-\frac {3986263268940827572255963529 x^7}{207094217017907652172185600}+\frac {21730712888356628741772337 x^6}{10920984100553723845017600}-\frac {1500040357444099007 x^5}{5129881450551705600}+\frac {4885269094757 x^4}{74217034874880}-\frac {2835642457 x^3}{108716359680}+\frac {11659 x^2}{524288}+\frac {15 x}{512}+\frac {3 i \sqrt {x}}{16}+1\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**4*Derivative(y(x), (x, 2)) + y(x)*sin(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
ValueError : ODE x**4*Derivative(y(x), (x, 2)) + y(x)*sin(x) does not match hint 2nd_power_series_regular