Internal problem ID [2419]
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: 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime } x +y^{\prime }-2 y x=x^{2}} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 53
Order:=6; dsolve(x*diff(y(x),x$2)+diff(y(x),x)-2*x*y(x)=x^2,y(x),type='series',x=0);
\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{2} x^{2}+\frac {1}{16} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+x^{3} \left (\frac {1}{9}+\frac {2}{225} x^{2}+\operatorname {O}\left (x^{3}\right )\right )+\left (-\frac {1}{2} x^{2}-\frac {3}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.05 (sec). Leaf size: 188
AsymptoticDSolveValue[x*y''[x]+y'[x]-2*x*y[x]==x^2,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^6}{288}+\frac {x^4}{16}+\frac {x^2}{2}+1\right )+c_1 \left (x^6 \left (\frac {\log (x)}{144}-\frac {1}{108}\right )+x^4 \left (\frac {\log (x)}{8}-\frac {1}{8}\right )+x^2 \left (\log (x)-\frac {1}{2}\right )+2 \log (x)+1\right )+\left (\frac {x^6}{288}+\frac {x^4}{16}+\frac {x^2}{2}+1\right ) \left (\frac {1}{100} x^5 (7-10 \log (x))+\frac {1}{18} x^3 (-6 \log (x)-1)\right )+\left (\frac {x^5}{20}+\frac {x^3}{6}\right ) \left (x^6 \left (\frac {\log (x)}{144}-\frac {1}{108}\right )+x^4 \left (\frac {\log (x)}{8}-\frac {1}{8}\right )+x^2 \left (\log (x)-\frac {1}{2}\right )+2 \log (x)+1\right ) \]