Internal problem ID [2417]
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: 1.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime } x +3 y^{\prime }-y=x} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 78
Order:=6; dsolve(x*diff(y(x),x$2)+3*diff(y(x),x)-y(x)=x,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} \left (1+\frac {1}{3} x +\frac {1}{24} x^{2}+\frac {1}{360} x^{3}+\frac {1}{8640} x^{4}+\frac {1}{302400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (\ln \left (x \right ) \left (x^{2}+\frac {1}{3} x^{3}+\frac {1}{24} x^{4}+\frac {1}{360} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+2 x -\frac {4}{9} x^{3}-\frac {25}{288} x^{4}-\frac {157}{21600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}}+x^{2} \left (\frac {1}{8}+\frac {1}{120} x +\frac {1}{2880} x^{2}+\frac {1}{100800} x^{3}+\operatorname {O}\left (x^{4}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.083 (sec). Leaf size: 248
AsymptoticDSolveValue[x*y''[x]+3*y'[x]-y[x]==x,y[x],{x,0,5}]
\[ y(x)\to \frac {c_2 \left (x^4 \left (\frac {25}{576}-\frac {\log (x)}{48}\right )+x^3 \left (\frac {2}{9}-\frac {\log (x)}{6}\right )-\frac {1}{2} x^2 \log (x)-x+1\right )}{x^2}+c_1 \left (\frac {x^5}{302400}+\frac {x^4}{8640}+\frac {x^3}{360}+\frac {x^2}{24}+\frac {x}{3}+1\right )+\left (\frac {x^5}{302400}+\frac {x^4}{8640}+\frac {x^3}{360}+\frac {x^2}{24}+\frac {x}{3}+1\right ) \left (\frac {x^6 (9-4 \log (x))}{2304}+\frac {1}{900} x^5 (23-15 \log (x))+\frac {1}{64} x^4 (1-4 \log (x))-\frac {x^3}{6}+\frac {x^2}{4}\right )+\frac {\left (-\frac {x^6}{288}-\frac {x^5}{30}-\frac {x^4}{8}\right ) \left (x^4 \left (\frac {25}{576}-\frac {\log (x)}{48}\right )+x^3 \left (\frac {2}{9}-\frac {\log (x)}{6}\right )-\frac {1}{2} x^2 \log (x)-x+1\right )}{x^2} \]