50.19.8 problem 2(d)
Internal
problem
ID
[8116]
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(d)
Date
solved
:
Monday, January 27, 2025 at 03:44:06 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{3} y^{\prime \prime }+\sin \left (x \right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Solution by Maple
Time used: 0.111 (sec). Leaf size: 162
Order:=8;
dsolve(x^3*diff(y(x),x$2)+sin(x)*y(x)=0,y(x),type='series',x=0);
\[
y = \sqrt {x}\, \left (c_{2} x^{\frac {i \sqrt {3}}{2}} \left (1+\frac {1}{12 i \sqrt {3}+24} x^{2}+\frac {1}{1440} \frac {-3 i \sqrt {3}-1}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} x^{4}+\frac {1}{362880} \frac {9 i \sqrt {3}-115}{\left (i \sqrt {3}+6\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{1} x^{-\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{12 i \sqrt {3}-24} x^{2}+\frac {-3 \sqrt {3}-i}{7200 i+8640 \sqrt {3}} x^{4}+\frac {9 \sqrt {3}-115 i}{4354560 i+14878080 \sqrt {3}} x^{6}+\operatorname {O}\left (x^{8}\right )\right )\right )
\]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 410
AsymptoticDSolveValue[x^3*D[y[x],{x,2}]+Sin[x]*y[x]==0,y[x],{x,0,"8"-1}]
\[
y(x)\to c_1 \left (\frac {\left (\frac {1}{5040}-\frac {1}{720 \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right )}+\frac {\frac {1}{36 \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right )}-\frac {1}{120}}{6 \left (1+\left (3-(-1)^{2/3}\right ) \left (4-(-1)^{2/3}\right )\right )}\right ) x^6}{1+\left (5-(-1)^{2/3}\right ) \left (6-(-1)^{2/3}\right )}+\frac {\left (\frac {1}{36 \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right )}-\frac {1}{120}\right ) x^4}{1+\left (3-(-1)^{2/3}\right ) \left (4-(-1)^{2/3}\right )}+\frac {x^2}{6 \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right )}+1\right ) x^{-(-1)^{2/3}}+c_2 \left (\frac {\left (\frac {1}{5040}-\frac {1}{720 \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right )}+\frac {\frac {1}{36 \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right )}-\frac {1}{120}}{6 \left (1+\left (3+\sqrt [3]{-1}\right ) \left (4+\sqrt [3]{-1}\right )\right )}\right ) x^6}{1+\left (5+\sqrt [3]{-1}\right ) \left (6+\sqrt [3]{-1}\right )}+\frac {\left (\frac {1}{36 \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right )}-\frac {1}{120}\right ) x^4}{1+\left (3+\sqrt [3]{-1}\right ) \left (4+\sqrt [3]{-1}\right )}+\frac {x^2}{6 \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right )}+1\right ) x^{\sqrt [3]{-1}}
\]