73.26.1 problem 36.2 (a)

Internal problem ID [15741]
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 (a)
Date solved : Tuesday, January 28, 2025 at 08:06:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-2 x 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: 33

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

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

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 44

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

\[ y(x)\to c_1 \left (\frac {x^5}{24}+\frac {x^3}{2}+x\right )+c_2 \left (\frac {x^6}{120}+\frac {x^4}{6}+x^2\right ) \]