Internal problem ID [2420]
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: 3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime } x -x y^{\prime }+y=x^{3}} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 54
Order:=6; dsolve(x*diff(y(x),x$2)-x*diff(y(x),x)+y(x)=x^3,y(x),type='series',x=0);
\[ y \left (x \right ) = x^{4} \left (\frac {1}{12}+\frac {1}{80} x +\operatorname {O}\left (x^{2}\right )\right )+\ln \left (x \right ) \left (-x +\operatorname {O}\left (x^{6}\right )\right ) c_{2} +c_{1} x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (1+x -\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {1}{72} x^{4}-\frac {1}{480} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.072 (sec). Leaf size: 139
AsymptoticDSolveValue[x*y''[x]-x*y'[x]+y[x]==x^3,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (-\frac {x^4}{72}-\frac {x^3}{12}-\frac {x^2}{2}-x \log (x)+1\right )+x \left (\frac {1}{36} x^6 (2-3 \log (x))+\frac {1}{25} x^5 (5 \log (x)-1)+\frac {1}{16} x^4 (-4 \log (x)-3)+\frac {x^3}{3}\right )+\left (-\frac {x^6}{12}+\frac {x^5}{5}-\frac {x^4}{4}\right ) \left (-\frac {x^4}{72}-\frac {x^3}{12}-\frac {x^2}{2}-x \log (x)+1\right )+c_2 x \]