73.26.5 problem 36.2 (e)

Internal problem ID [15745]
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 (e)
Date solved : Tuesday, January 28, 2025 at 08:06:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.067 (sec). Leaf size: 48

Order:=6; 
dsolve(x^2*diff(y(x),x$2)-(5*x+2*x^2)*diff(y(x),x)+9*y(x)=0,y(x),type='series',x=0);
 

\[ y = x^{3} \left (\left (\ln \left (x \right ) c_{2} +c_{1} \right ) \left (1+6 x +12 x^{2}+\frac {40}{3} x^{3}+10 x^{4}+\frac {28}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-10\right ) x -29 x^{2}-\frac {346}{9} x^{3}-\frac {193}{6} x^{4}-\frac {1459}{75} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 112

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

\[ y(x)\to c_1 \left (\frac {28 x^5}{5}+10 x^4+\frac {40 x^3}{3}+12 x^2+6 x+1\right ) x^3+c_2 \left (\left (-\frac {1459 x^5}{75}-\frac {193 x^4}{6}-\frac {346 x^3}{9}-29 x^2-10 x\right ) x^3+\left (\frac {28 x^5}{5}+10 x^4+\frac {40 x^3}{3}+12 x^2+6 x+1\right ) x^3 \log (x)\right ) \]