Internal problem ID [14012]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 36. The big theorem on the the Frobenius method. Additional Exercises. page
739
Problem number: 36.2 (h).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -\left (2 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 63
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)-(1+2*x)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1+\frac {2}{3} x +\frac {1}{6} x^{2}+\frac {1}{45} x^{3}+\frac {1}{540} x^{4}+\frac {1}{9450} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (4 x^{2}+\frac {8}{3} x^{3}+\frac {2}{3} x^{4}+\frac {4}{45} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+4 x -\frac {32}{9} x^{3}-\frac {25}{18} x^{4}-\frac {157}{675} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.02 (sec). Leaf size: 83
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]-(1+2*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {31 x^4+88 x^3+36 x^2-72 x+36}{36 x}-\frac {1}{3} x \left (x^2+4 x+6\right ) \log (x)\right )+c_2 \left (\frac {x^5}{540}+\frac {x^4}{45}+\frac {x^3}{6}+\frac {2 x^2}{3}+x\right ) \]