50.22.15 problem 2(g)
Internal
problem
ID
[8158]
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
:
2(g)
Date
solved
:
Monday, January 27, 2025 at 03:45:00 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }+y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Solution by Maple
Time used: 0.012 (sec). Leaf size: 55
Order:=8;
dsolve(x^2*diff(y(x),x$2)+x*(1-x)*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[
y = c_{1} x^{-i} \left (1+\left (\frac {2}{5}-\frac {i}{5}\right ) x +\left (\frac {1}{10}-\frac {i}{20}\right ) x^{2}+\left (\frac {17}{780}-\frac {i}{130}\right ) x^{3}+\left (\frac {5}{1248}-\frac {i}{1248}\right ) x^{4}+\left (\frac {113}{180960}-\frac {7 i}{180960}\right ) x^{5}+\left (\frac {911}{10857600}+\frac {19 i}{3619200}\right ) x^{6}+\left (\frac {39799}{4028169600}+\frac {1009 i}{575452800}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{i} \left (1+\left (\frac {2}{5}+\frac {i}{5}\right ) x +\left (\frac {1}{10}+\frac {i}{20}\right ) x^{2}+\left (\frac {17}{780}+\frac {i}{130}\right ) x^{3}+\left (\frac {5}{1248}+\frac {i}{1248}\right ) x^{4}+\left (\frac {113}{180960}+\frac {7 i}{180960}\right ) x^{5}+\left (\frac {911}{10857600}-\frac {19 i}{3619200}\right ) x^{6}+\left (\frac {39799}{4028169600}-\frac {1009 i}{575452800}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right )
\]
✓ Solution by Mathematica
Time used: 0.020 (sec). Leaf size: 122
AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+x*(1-x)*D[y[x],x]+y[x]==0,y[x],{x,0,"8"-1}]
\[
y(x)\to \left (\frac {59}{10857600}-\frac {17 i}{10857600}\right ) c_2 x^{-i} \left ((14+5 i) x^6+(108+24 i) x^5+(720+60 i) x^4+(4080-240 i) x^3+(19440-3600 i) x^2+(77760-14400 i) x+(169920+48960 i)\right )+\left (\frac {59}{10857600}+\frac {17 i}{10857600}\right ) c_1 x^i \left ((14-5 i) x^6+(108-24 i) x^5+(720-60 i) x^4+(4080+240 i) x^3+(19440+3600 i) x^2+(77760+14400 i) x+(169920-48960 i)\right )
\]