20.25.11 problem 12
Internal
problem
ID
[4016]
Book
:
Differential
equations
and
linear
algebra,
Stephen
W.
Goode
and
Scott
A
Annin.
Fourth
edition,
2015
Section
:
Chapter
11,
Series
Solutions
to
Linear
Differential
Equations.
Exercises
for
11.4.
page
758
Problem
number
:
12
Date
solved
:
Monday, January 27, 2025 at 08:06:17 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }-\left (x +2\right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Solution by Maple
Time used: 0.014 (sec). Leaf size: 169
Order:=6;
dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)-(2+x)*y(x)=0,y(x),type='series',x=0);
\[
y \left (x \right ) = c_{1} x^{-\sqrt {2}} \left (1+\frac {1}{1-2 \sqrt {2}} x +\frac {1}{20-12 \sqrt {2}} x^{2}-\frac {1}{228 \sqrt {2}-324} x^{3}+\frac {1}{8832-6240 \sqrt {2}} x^{4}-\frac {1}{480} \frac {1}{\left (2 \sqrt {2}-1\right ) \left (\sqrt {2}-1\right ) \left (-3+2 \sqrt {2}\right ) \left (-2+\sqrt {2}\right ) \left (-5+2 \sqrt {2}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\sqrt {2}} \left (1+\frac {1}{1+2 \sqrt {2}} x +\frac {1}{20+12 \sqrt {2}} x^{2}+\frac {1}{228 \sqrt {2}+324} x^{3}+\frac {1}{8832+6240 \sqrt {2}} x^{4}+\frac {1}{244320 \sqrt {2}+345600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )
\]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 843
AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+x*D[y[x],x]-(2+x)*y[x]==0,y[x],{x,0,"6"-1}]
\[
y(x)\to \left (\frac {x^5}{\left (-1+\sqrt {2}+\sqrt {2} \left (1+\sqrt {2}\right )\right ) \left (\sqrt {2}+\left (1+\sqrt {2}\right ) \left (2+\sqrt {2}\right )\right ) \left (1+\sqrt {2}+\left (2+\sqrt {2}\right ) \left (3+\sqrt {2}\right )\right ) \left (2+\sqrt {2}+\left (3+\sqrt {2}\right ) \left (4+\sqrt {2}\right )\right ) \left (3+\sqrt {2}+\left (4+\sqrt {2}\right ) \left (5+\sqrt {2}\right )\right )}+\frac {x^4}{\left (-1+\sqrt {2}+\sqrt {2} \left (1+\sqrt {2}\right )\right ) \left (\sqrt {2}+\left (1+\sqrt {2}\right ) \left (2+\sqrt {2}\right )\right ) \left (1+\sqrt {2}+\left (2+\sqrt {2}\right ) \left (3+\sqrt {2}\right )\right ) \left (2+\sqrt {2}+\left (3+\sqrt {2}\right ) \left (4+\sqrt {2}\right )\right )}+\frac {x^3}{\left (-1+\sqrt {2}+\sqrt {2} \left (1+\sqrt {2}\right )\right ) \left (\sqrt {2}+\left (1+\sqrt {2}\right ) \left (2+\sqrt {2}\right )\right ) \left (1+\sqrt {2}+\left (2+\sqrt {2}\right ) \left (3+\sqrt {2}\right )\right )}+\frac {x^2}{\left (-1+\sqrt {2}+\sqrt {2} \left (1+\sqrt {2}\right )\right ) \left (\sqrt {2}+\left (1+\sqrt {2}\right ) \left (2+\sqrt {2}\right )\right )}+\frac {x}{-1+\sqrt {2}+\sqrt {2} \left (1+\sqrt {2}\right )}+1\right ) c_1 x^{\sqrt {2}}+\left (\frac {x^5}{\left (-1-\sqrt {2}-\sqrt {2} \left (1-\sqrt {2}\right )\right ) \left (-\sqrt {2}+\left (1-\sqrt {2}\right ) \left (2-\sqrt {2}\right )\right ) \left (1-\sqrt {2}+\left (2-\sqrt {2}\right ) \left (3-\sqrt {2}\right )\right ) \left (2-\sqrt {2}+\left (3-\sqrt {2}\right ) \left (4-\sqrt {2}\right )\right ) \left (3-\sqrt {2}+\left (4-\sqrt {2}\right ) \left (5-\sqrt {2}\right )\right )}+\frac {x^4}{\left (-1-\sqrt {2}-\sqrt {2} \left (1-\sqrt {2}\right )\right ) \left (-\sqrt {2}+\left (1-\sqrt {2}\right ) \left (2-\sqrt {2}\right )\right ) \left (1-\sqrt {2}+\left (2-\sqrt {2}\right ) \left (3-\sqrt {2}\right )\right ) \left (2-\sqrt {2}+\left (3-\sqrt {2}\right ) \left (4-\sqrt {2}\right )\right )}+\frac {x^3}{\left (-1-\sqrt {2}-\sqrt {2} \left (1-\sqrt {2}\right )\right ) \left (-\sqrt {2}+\left (1-\sqrt {2}\right ) \left (2-\sqrt {2}\right )\right ) \left (1-\sqrt {2}+\left (2-\sqrt {2}\right ) \left (3-\sqrt {2}\right )\right )}+\frac {x^2}{\left (-1-\sqrt {2}-\sqrt {2} \left (1-\sqrt {2}\right )\right ) \left (-\sqrt {2}+\left (1-\sqrt {2}\right ) \left (2-\sqrt {2}\right )\right )}+\frac {x}{-1-\sqrt {2}-\sqrt {2} \left (1-\sqrt {2}\right )}+1\right ) c_2 x^{-\sqrt {2}}
\]