7.13.23 problem 23
Internal
problem
ID
[422]
Book
:
Elementary
Differential
Equations.
By
C.
Henry
Edwards,
David
E.
Penney
and
David
Calvis.
6th
edition.
2008
Section
:
Chapter
3.
Power
series
methods.
Section
3.1
(Introduction).
Problems
at
page
206
Problem
number
:
23
Date
solved
:
Wednesday, February 05, 2025 at 03:42:13 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{2} y^{\prime \prime }+x^{2} y^{\prime }+y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 228
Order:=6;
dsolve(x^2*diff(y(x),x$2)+x^2*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[
y = \sqrt {x}\, \left (c_2 \,x^{\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{2} x +\frac {i \sqrt {3}+3}{8 i \sqrt {3}+16} x^{2}+\frac {-i \sqrt {3}-5}{48 i \sqrt {3}+96} x^{3}+\frac {1}{384} \frac {\left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+7\right )}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} x^{4}-\frac {1}{3840} \frac {\left (i \sqrt {3}+7\right ) \left (i \sqrt {3}+9\right )}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \,x^{-\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{2} x +\frac {\sqrt {3}+3 i}{8 \sqrt {3}+16 i} x^{2}+\frac {-\sqrt {3}-5 i}{48 \sqrt {3}+96 i} x^{3}+\frac {3 i \sqrt {3}-8}{576 i \sqrt {3}-480} x^{4}-\frac {1}{3840} \frac {\left (\sqrt {3}+7 i\right ) \left (\sqrt {3}+9 i\right )}{\left (\sqrt {3}+4 i\right ) \left (\sqrt {3}+2 i\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )
\]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 886
AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+x^2*D[y[x],x]+y[x]==0,y[x],{x,0,"6"-1}]
\[
y(x)\to \left (\frac {(-1)^{2/3} \left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right ) \left (3-(-1)^{2/3}\right ) \left (4-(-1)^{2/3}\right ) x^5}{\left (1-(-1)^{2/3} \left (1-(-1)^{2/3}\right )\right ) \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right ) \left (1+\left (2-(-1)^{2/3}\right ) \left (3-(-1)^{2/3}\right )\right ) \left (1+\left (3-(-1)^{2/3}\right ) \left (4-(-1)^{2/3}\right )\right ) \left (1+\left (4-(-1)^{2/3}\right ) \left (5-(-1)^{2/3}\right )\right )}-\frac {(-1)^{2/3} \left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right ) \left (3-(-1)^{2/3}\right ) x^4}{\left (1-(-1)^{2/3} \left (1-(-1)^{2/3}\right )\right ) \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right ) \left (1+\left (2-(-1)^{2/3}\right ) \left (3-(-1)^{2/3}\right )\right ) \left (1+\left (3-(-1)^{2/3}\right ) \left (4-(-1)^{2/3}\right )\right )}+\frac {(-1)^{2/3} \left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right ) x^3}{\left (1-(-1)^{2/3} \left (1-(-1)^{2/3}\right )\right ) \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right ) \left (1+\left (2-(-1)^{2/3}\right ) \left (3-(-1)^{2/3}\right )\right )}-\frac {(-1)^{2/3} \left (1-(-1)^{2/3}\right ) x^2}{\left (1-(-1)^{2/3} \left (1-(-1)^{2/3}\right )\right ) \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right )}+\frac {(-1)^{2/3} x}{1-(-1)^{2/3} \left (1-(-1)^{2/3}\right )}+1\right ) c_1 x^{-(-1)^{2/3}}+\left (-\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right ) \left (3+\sqrt [3]{-1}\right ) \left (4+\sqrt [3]{-1}\right ) x^5}{\left (1+\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right )\right ) \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right ) \left (1+\left (2+\sqrt [3]{-1}\right ) \left (3+\sqrt [3]{-1}\right )\right ) \left (1+\left (3+\sqrt [3]{-1}\right ) \left (4+\sqrt [3]{-1}\right )\right ) \left (1+\left (4+\sqrt [3]{-1}\right ) \left (5+\sqrt [3]{-1}\right )\right )}+\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right ) \left (3+\sqrt [3]{-1}\right ) x^4}{\left (1+\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right )\right ) \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right ) \left (1+\left (2+\sqrt [3]{-1}\right ) \left (3+\sqrt [3]{-1}\right )\right ) \left (1+\left (3+\sqrt [3]{-1}\right ) \left (4+\sqrt [3]{-1}\right )\right )}-\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right ) x^3}{\left (1+\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right )\right ) \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right ) \left (1+\left (2+\sqrt [3]{-1}\right ) \left (3+\sqrt [3]{-1}\right )\right )}+\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) x^2}{\left (1+\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right )\right ) \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right )}-\frac {\sqrt [3]{-1} x}{1+\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right )}+1\right ) c_2 x^{\sqrt [3]{-1}}
\]