50.22.17 problem 3(a)
Internal
problem
ID
[8160]
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
:
3(a)
Date
solved
:
Sunday, March 30, 2025 at 12:47:02 PM
CAS
classification
:
[[_3rd_order, _with_linear_symmetries]]
\begin{align*} x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+x y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.038 (sec). Leaf size: 476
Order:=8;
ode:=x^3*diff(diff(diff(y(x),x),x),x)+2*x^2*diff(diff(y(x),x),x)+(x^2+x)*diff(y(x),x)+x*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = c_1 \sqrt {x}\, x^{-\frac {i \sqrt {3}}{2}} \left (1+\frac {1}{-1+i \sqrt {3}} x -\frac {1}{2+6 i \sqrt {3}} x^{2}-\frac {1}{18} \frac {1}{\left (\sqrt {3}+i\right ) \left (\frac {\sqrt {3}}{3}+i\right ) \left (i \sqrt {3}-2\right )} x^{3}+\frac {1}{-1728-480 i \sqrt {3}} x^{4}+\frac {1}{3360 i \sqrt {3}+50400} x^{5}-\frac {1}{720} \frac {1}{\left (6 i+\sqrt {3}\right ) \left (\sqrt {3}+5 i\right ) \left (4 i+\sqrt {3}\right ) \left (\sqrt {3}+3 i\right ) \left (2 i+\sqrt {3}\right ) \left (\sqrt {3}+i\right )} x^{6}-\frac {1}{5040} \frac {1}{\left (i \sqrt {3}-2\right ) \left (\sqrt {3}+7 i\right ) \left (6 i+\sqrt {3}\right ) \left (\sqrt {3}+5 i\right ) \left (4 i+\sqrt {3}\right ) \left (\sqrt {3}+3 i\right ) \left (\sqrt {3}+i\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \sqrt {x}\, x^{\frac {i \sqrt {3}}{2}} \left (1+\frac {1}{-1-i \sqrt {3}} x +\frac {1}{6 i \sqrt {3}-2} x^{2}+\frac {1}{6} \frac {1}{\left (-3 i+\sqrt {3}\right ) \left (-i+\sqrt {3}\right ) \left (i \sqrt {3}+2\right )} x^{3}+\frac {1}{24} \frac {1}{\left (3 i-\sqrt {3}\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right ) \left (-i+\sqrt {3}\right )} x^{4}+\frac {1}{120} \frac {1}{\left (-3 i+\sqrt {3}\right ) \left (-i+\sqrt {3}\right ) \left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} x^{5}+\frac {1}{720} \frac {1}{\left (3 i-\sqrt {3}\right ) \left (i \sqrt {3}+6\right ) \left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right ) \left (-i+\sqrt {3}\right )} x^{6}+\frac {1}{5040} \frac {1}{\left (-3 i+\sqrt {3}\right ) \left (-i+\sqrt {3}\right ) \left (i \sqrt {3}+7\right ) \left (i \sqrt {3}+6\right ) \left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_3 \left (1-x +\frac {1}{3} x^{2}-\frac {1}{21} x^{3}+\frac {1}{273} x^{4}-\frac {1}{5733} x^{5}+\frac {1}{177723} x^{6}-\frac {1}{7642089} x^{7}+\operatorname {O}\left (x^{8}\right )\right )
\]
✓ Mathematica. Time used: 0.01 (sec). Leaf size: 3447
ode=x^3*D[y[x],{x,3}]+2*x^2*D[y[x],{x,2}]+(x+x^2)*D[y[x],x]+x*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
Too large to display
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(1 - x)*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
Series solution not supported for ode of order > 2