15.25.1 problem 1
Internal
problem
ID
[3388]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
43,
page
209
Problem
number
:
1
Date
solved
:
Sunday, March 30, 2025 at 01:38:48 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x y^{\prime \prime }+3 y^{\prime }-y&=x \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.024 (sec). Leaf size: 74
Order:=6;
ode:=x*diff(diff(y(x),x),x)+3*diff(y(x),x)-y(x) = x;
dsolve(ode,y(x),type='series',x=0);
\[
y = c_1 \left (1+\frac {1}{3} x +\frac {1}{24} x^{2}+\frac {1}{360} x^{3}+\frac {1}{8640} x^{4}+\frac {1}{302400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (\ln \left (x \right ) \left (x^{2}+\frac {1}{3} x^{3}+\frac {1}{24} x^{4}+\frac {1}{360} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+2 x -\frac {4}{9} x^{3}-\frac {25}{288} x^{4}-\frac {157}{21600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}}+x^{2} \left (\frac {1}{8}+\frac {1}{120} x +\frac {1}{2880} x^{2}+\frac {1}{100800} x^{3}+\operatorname {O}\left (x^{4}\right )\right )
\]
✓ Mathematica. Time used: 0.086 (sec). Leaf size: 248
ode=x*D[y[x],{x,2}]+3*D[y[x],x]-y[x]==x;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
\[
y(x)\to \frac {c_2 \left (x^4 \left (\frac {25}{576}-\frac {\log (x)}{48}\right )+x^3 \left (\frac {2}{9}-\frac {\log (x)}{6}\right )-\frac {1}{2} x^2 \log (x)-x+1\right )}{x^2}+c_1 \left (\frac {x^5}{302400}+\frac {x^4}{8640}+\frac {x^3}{360}+\frac {x^2}{24}+\frac {x}{3}+1\right )+\left (\frac {x^5}{302400}+\frac {x^4}{8640}+\frac {x^3}{360}+\frac {x^2}{24}+\frac {x}{3}+1\right ) \left (\frac {x^6 (9-4 \log (x))}{2304}+\frac {1}{900} x^5 (23-15 \log (x))+\frac {1}{64} x^4 (1-4 \log (x))-\frac {x^3}{6}+\frac {x^2}{4}\right )+\frac {\left (-\frac {x^6}{288}-\frac {x^5}{30}-\frac {x^4}{8}\right ) \left (x^4 \left (\frac {25}{576}-\frac {\log (x)}{48}\right )+x^3 \left (\frac {2}{9}-\frac {\log (x)}{6}\right )-\frac {1}{2} x^2 \log (x)-x+1\right )}{x^2}
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*Derivative(y(x), (x, 2)) - x - y(x) + 3*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
ValueError : ODE x*Derivative(y(x), (x, 2)) - x - y(x) + 3*Derivative(y(x), x) does not match hint 2nd_power_series_regular