15.25.7 problem 6
Internal
problem
ID
[3394]
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
:
6
Date
solved
:
Monday, January 27, 2025 at 07:35:36 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (x^{2}+3\right ) y^{\prime }+y&=-2 x^{2}+x \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 64
Order:=6;
dsolve(x^2*(x+1)*diff(y(x),x$2)+x*(x^2+3)*diff(y(x),x)+y(x)=x-2*x^2,y(x),type='series',x=0);
\[
y \left (x \right ) = \frac {x^{2} \left (\frac {1}{4}-\frac {2}{9} x +\frac {7}{576} x^{2}+\frac {107}{7200} x^{3}-\frac {1031}{172800} x^{4}+\operatorname {O}\left (x^{5}\right )\right )+\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-2 x +\frac {1}{4} x^{2}-\frac {1}{64} x^{4}+\frac {3}{800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (7 x -x^{2}+\frac {7}{36} x^{3}+\frac {35}{1152} x^{4}-\frac {191}{9000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{x}
\]
✓ Solution by Mathematica
Time used: 0.139 (sec). Leaf size: 300
AsymptoticDSolveValue[x^2*(x+1)*D[y[x],{x,2}]+x*(x^2+3)*D[y[x],x]+y[x]==x-2*x^2,y[x],{x,0,"6"-1}]
\[
y(x)\to \frac {c_2 \left (\frac {3 x^5}{800}-\frac {x^4}{64}+\frac {x^2}{4}-2 x+1\right )}{x}+\frac {c_1 \left (x^5 \left (\frac {3 \log (x)}{800}-\frac {629}{36000}\right )+x^4 \left (\frac {17}{1152}-\frac {\log (x)}{64}\right )+\frac {7 x^3}{36}+x^2 \left (\frac {\log (x)}{4}-\frac {3}{4}\right )+x (5-2 \log (x))+\log (x)+1\right )}{x}+\frac {\left (\frac {34109 x^6}{1152}-\frac {491 x^5}{30}+\frac {123 x^4}{16}-\frac {8 x^3}{3}+\frac {x^2}{2}\right ) \left (x^5 \left (\frac {3 \log (x)}{800}-\frac {629}{36000}\right )+x^4 \left (\frac {17}{1152}-\frac {\log (x)}{64}\right )+\frac {7 x^3}{36}+x^2 \left (\frac {\log (x)}{4}-\frac {3}{4}\right )+x (5-2 \log (x))+\log (x)+1\right )}{x}+\frac {\left (\frac {3 x^5}{800}-\frac {x^4}{64}+\frac {x^2}{4}-2 x+1\right ) \left (\frac {x^6 (33586-34109 \log (x))}{1152}+\frac {1}{900} x^5 (14730 \log (x)-12641)+\frac {1}{64} x^4 (319-492 \log (x))+\frac {1}{9} x^3 (24 \log (x)-5)+\frac {1}{4} x^2 (-2 \log (x)-1)\right )}{x}
\]