56.4.43 problem 40
Internal
problem
ID
[8932]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
4.0
Problem
number
:
40
Date
solved
:
Sunday, March 30, 2025 at 01:55:40 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} 2 x^{2} y^{\prime \prime }+2 x y^{\prime }-x y&=\sin \left (x \right ) \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.027 (sec). Leaf size: 62
Order:=6;
ode:=2*x^2*diff(diff(y(x),x),x)+2*x*diff(y(x),x)-x*y(x) = sin(x);
dsolve(ode,y(x),type='series',x=0);
\[
y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\frac {1}{2} x +\frac {1}{16} x^{2}+\frac {1}{288} x^{3}+\frac {1}{9216} x^{4}+\frac {1}{460800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+x \left (\frac {1}{2}+\frac {1}{16} x -\frac {5}{864} x^{2}-\frac {5}{27648} x^{3}+\frac {1127}{6912000} x^{4}+\frac {1127}{497664000} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-x -\frac {3}{16} x^{2}-\frac {11}{864} x^{3}-\frac {25}{55296} x^{4}-\frac {137}{13824000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2
\]
✓ Mathematica. Time used: 0.154 (sec). Leaf size: 340
ode=2*x^2*D[y[x],{x,2}]+2*x*D[y[x],x]-x*y[x]==Sin[x];
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
\[
y(x)\to c_2 \left (\frac {x^5}{460800}+\frac {x^4}{9216}+\frac {x^3}{288}+\frac {x^2}{16}+\frac {x}{2}+1\right )+c_1 \left (x^5 \left (\frac {\log (x)}{460800}-\frac {107}{13824000}\right )+x^4 \left (\frac {\log (x)}{9216}-\frac {19}{55296}\right )+x^3 \left (\frac {\log (x)}{288}-\frac {1}{108}\right )+x^2 \left (\frac {\log (x)}{16}-\frac {1}{8}\right )+x \left (\frac {\log (x)}{2}-\frac {1}{2}\right )+\log (x)+1\right )+\left (\frac {4963 x^6}{16588800}-\frac {91 x^5}{460800}-\frac {23 x^4}{2304}-\frac {5 x^3}{288}+\frac {x^2}{8}+\frac {x}{2}\right ) \left (x^5 \left (\frac {\log (x)}{460800}-\frac {107}{13824000}\right )+x^4 \left (\frac {\log (x)}{9216}-\frac {19}{55296}\right )+x^3 \left (\frac {\log (x)}{288}-\frac {1}{108}\right )+x^2 \left (\frac {\log (x)}{16}-\frac {1}{8}\right )+x \left (\frac {\log (x)}{2}-\frac {1}{2}\right )+\log (x)+1\right )+\left (\frac {x^5}{460800}+\frac {x^4}{9216}+\frac {x^3}{288}+\frac {x^2}{16}+\frac {x}{2}+1\right ) \left (\frac {x^6 (66968-74445 \log (x))}{248832000}+\frac {13 x^5 (210 \log (x)-3107)}{13824000}+\frac {x^4 (276 \log (x)-325)}{27648}+\frac {1}{864} x^3 (15 \log (x)+37)+\frac {1}{16} x^2 (3-2 \log (x))-\frac {1}{2} x \log (x)\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x**2*Derivative(y(x), (x, 2)) - x*y(x) + 2*x*Derivative(y(x), x) - sin(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
ValueError : ODE 2*x**2*Derivative(y(x), (x, 2)) - x*y(x) + 2*x*Derivative(y(x), x) - sin(x) does not match hint 2nd_power_series_regular