56.4.44 problem 41
Internal
problem
ID
[8933]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
4.0
Problem
number
:
41
Date
solved
:
Monday, January 27, 2025 at 05:20:17 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} 2 x^{2} y^{\prime \prime }+2 x y^{\prime }-y x&=x \sin \left (x \right ) \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Solution by Maple
Time used: 0.013 (sec). Leaf size: 60
Order:=6;
dsolve(2*x^2*diff(y(x), x, x) + 2*x*diff(y(x), x) - x*y(x) = x*sin(x),y(x),type='series',x=0);
\[
y = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{2} x +\frac {1}{16} x^{2}+\frac {1}{288} x^{3}+\frac {1}{9216} x^{4}+\frac {1}{460800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+x^{2} \left (\frac {1}{8}+\frac {1}{144} x -\frac {23}{4608} x^{2}-\frac {23}{230400} x^{3}+\operatorname {O}\left (x^{4}\right )\right )+\left (-x -\frac {3}{16} x^{2}-\frac {11}{864} x^{3}-\frac {25}{55296} x^{4}-\frac {137}{13824000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}
\]
✓ Solution by Mathematica
Time used: 0.167 (sec). Leaf size: 328
AsymptoticDSolveValue[2*x^2*D[y[x],{x,2}]+2*x*D[y[x],x]-x*y[x]==x*Sin[x],y[x],{x,0,"6"-1}]
\[
y(x)\to c_2 \left (\frac {x^5}{460800}+\frac {x^4}{9216}+\frac {x^3}{288}+\frac {x^2}{16}+\frac {x}{2}+1\right )+c_1 \left (x^5 \left (\frac {\log (x)}{460800}-\frac {107}{13824000}\right )+x^4 \left (\frac {\log (x)}{9216}-\frac {19}{55296}\right )+x^3 \left (\frac {\log (x)}{288}-\frac {1}{108}\right )+x^2 \left (\frac {\log (x)}{16}-\frac {1}{8}\right )+x \left (\frac {\log (x)}{2}-\frac {1}{2}\right )+\log (x)+1\right )+\left (-\frac {91 x^6}{552960}-\frac {23 x^5}{2880}-\frac {5 x^4}{384}+\frac {x^3}{12}+\frac {x^2}{4}\right ) \left (x^5 \left (\frac {\log (x)}{460800}-\frac {107}{13824000}\right )+x^4 \left (\frac {\log (x)}{9216}-\frac {19}{55296}\right )+x^3 \left (\frac {\log (x)}{288}-\frac {1}{108}\right )+x^2 \left (\frac {\log (x)}{16}-\frac {1}{8}\right )+x \left (\frac {\log (x)}{2}-\frac {1}{2}\right )+\log (x)+1\right )+\left (\frac {x^5}{460800}+\frac {x^4}{9216}+\frac {x^3}{288}+\frac {x^2}{16}+\frac {x}{2}+1\right ) \left (\frac {13 x^6 (21 \log (x)-310)}{1658880}+\frac {x^5 (345 \log (x)-389)}{43200}+\frac {x^4 (20 \log (x)+51)}{1536}+\frac {1}{36} x^3 (4-3 \log (x))+\frac {1}{8} x^2 (-2 \log (x)-1)\right )
\]