Internal problem ID [5464]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 26. Integration in series (singular points). Supplemetary problems. Page
218
Problem number: 16.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
\[ \boxed {x y^{\prime \prime }-2 y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 56
Order:=6; dsolve(x*diff(y(x),x$2)-2*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y = c_{1} x^{3} \left (1-\frac {1}{4} x +\frac {1}{40} x^{2}-\frac {1}{720} x^{3}+\frac {1}{20160} x^{4}-\frac {1}{806400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x^{3}+\frac {1}{4} x^{4}-\frac {1}{40} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+6 x +3 x^{2}-\frac {5}{16} x^{4}+\frac {39}{800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.018 (sec). Leaf size: 79
AsymptoticDSolveValue[x*y''[x]-2*y'[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {1}{48} (x-4) x^3 \log (x)+\frac {1}{576} \left (-19 x^4+16 x^3+144 x^2+288 x+576\right )\right )+c_2 \left (\frac {x^7}{20160}-\frac {x^6}{720}+\frac {x^5}{40}-\frac {x^4}{4}+x^3\right ) \]