Internal problem ID [13995]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional
Exercises. page 715
Problem number: 35.4 (i).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }-\left (2 x^{2}+5 x \right ) y^{\prime }+\left (9+4 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 69
Order:=6; dsolve(x^2*diff(y(x),x$2)-(5*x+2*x^2)*diff(y(x),x)+(9+4*x)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \left (\left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1+2 x +2 x^{2}+\frac {4}{3} x^{3}+\frac {2}{3} x^{4}+\frac {4}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-2\right ) x -3 x^{2}-\frac {22}{9} x^{3}-\frac {25}{18} x^{4}-\frac {137}{225} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x^{3} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 116
AsymptoticDSolveValue[x^2*y''[x]-(5*x+2*x^2)*y'[x]+(9+4*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {4 x^5}{15}+\frac {2 x^4}{3}+\frac {4 x^3}{3}+2 x^2+2 x+1\right ) x^3+c_2 \left (\left (-\frac {137 x^5}{225}-\frac {25 x^4}{18}-\frac {22 x^3}{9}-3 x^2-2 x\right ) x^3+\left (\frac {4 x^5}{15}+\frac {2 x^4}{3}+\frac {4 x^3}{3}+2 x^2+2 x+1\right ) x^3 \log (x)\right ) \]