50.22.10 problem 2(b)
Internal
problem
ID
[8153]
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(b)
Date
solved
:
Monday, January 27, 2025 at 03:44:54 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+y \left (1+x \right )&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Solution by Maple
Time used: 0.013 (sec). Leaf size: 55
Order:=8;
dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(1+x)*y(x)=0,y(x),type='series',x=0);
\[
y = c_{1} x^{-i} \left (1+\left (-\frac {1}{5}-\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}+\frac {3 i}{40}\right ) x^{2}+\left (\frac {3}{520}-\frac {7 i}{1560}\right ) x^{3}+\left (-\frac {1}{2496}+\frac {i}{12480}\right ) x^{4}+\left (\frac {9}{603200}+\frac {i}{361920}\right ) x^{5}+\left (-\frac {19}{54288000}-\frac {7 i}{36192000}\right ) x^{6}+\left (\frac {1}{179829000}+\frac {223 i}{40281696000}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{i} \left (1+\left (-\frac {1}{5}+\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}-\frac {3 i}{40}\right ) x^{2}+\left (\frac {3}{520}+\frac {7 i}{1560}\right ) x^{3}+\left (-\frac {1}{2496}-\frac {i}{12480}\right ) x^{4}+\left (\frac {9}{603200}-\frac {i}{361920}\right ) x^{5}+\left (-\frac {19}{54288000}+\frac {7 i}{36192000}\right ) x^{6}+\left (\frac {1}{179829000}-\frac {223 i}{40281696000}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right )
\]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 118
AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+x*D[y[x],x]+(1+x)*y[x]==0,y[x],{x,0,"8"-1}]
\[
y(x)\to \left (\frac {7}{36192000}+\frac {19 i}{54288000}\right ) c_1 x^i \left (i x^6+(12-36 i) x^5-(660-780 i) x^4+(16800-7200 i) x^3-(194400+36000 i) x^2+(633600+921600 i) x+(1209600-2188800 i)\right )-\left (\frac {19}{54288000}+\frac {7 i}{36192000}\right ) c_2 x^{-i} \left (x^6-(36-12 i) x^5+(780-660 i) x^4-(7200-16800 i) x^3-(36000+194400 i) x^2+(921600+633600 i) x-(2188800-1209600 i)\right )
\]