73.25.23 problem 35.4 (i)

Internal problem ID [15731]
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)
Date solved : Tuesday, January 28, 2025 at 08:06:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-\left (2 x^{2}+5 x \right ) y^{\prime }+\left (9+4 x \right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 48

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);
 

\[ y = x^{3} \left (\left (\ln \left (x \right ) c_{2} +c_{1} \right ) \left (1+2 x +2 x^{2}+\frac {4}{3} x^{3}+\frac {2}{3} x^{4}+\frac {4}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-2\right ) x -3 x^{2}-\frac {22}{9} x^{3}-\frac {25}{18} x^{4}-\frac {137}{225} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 116

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]-(5*x+2*x^2)*D[y[x],x]+(9+4*x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {4 x^5}{15}+\frac {2 x^4}{3}+\frac {4 x^3}{3}+2 x^2+2 x+1\right ) x^3+c_2 \left (\left (-\frac {137 x^5}{225}-\frac {25 x^4}{18}-\frac {22 x^3}{9}-3 x^2-2 x\right ) x^3+\left (\frac {4 x^5}{15}+\frac {2 x^4}{3}+\frac {4 x^3}{3}+2 x^2+2 x+1\right ) x^3 \log (x)\right ) \]