73.26.2 problem 36.2 (b)

Internal problem ID [15742]
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 (b)
Date solved : Tuesday, January 28, 2025 at 08:06:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.065 (sec). Leaf size: 47

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

\[ y = c_{1} x^{2} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12+12 x +6 x^{2}+4 x^{3}+\frac {5}{2} x^{4}+\frac {11}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.026 (sec). Leaf size: 62

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

\[ y(x)\to c_1 \left (\frac {5 x^3}{24}+\frac {x^2}{3}+\frac {x}{2}+\frac {1}{x}+1\right )+c_2 \left (\frac {x^6}{24}+\frac {x^5}{6}+\frac {x^4}{2}+x^3+x^2\right ) \]