15.24.9 problem 9

Internal problem ID [3381]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 42, page 206
Problem number : 9
Date solved : Monday, January 27, 2025 at 07:35:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-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: 0.016 (sec). Leaf size: 47

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

\[ y \left (x \right ) = c_{1} x^{2} \left (1+\frac {1}{2} x +\frac {1}{20} x^{2}-\frac {1}{60} x^{3}-\frac {1}{210} x^{4}-\frac {1}{3360} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12+6 x +6 x^{2}+5 x^{3}+x^{4}-\frac {1}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.026 (sec). Leaf size: 68

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

\[ y(x)\to c_1 \left (\frac {x^3}{12}+\frac {5 x^2}{12}+\frac {x}{2}+\frac {1}{x}+\frac {1}{2}\right )+c_2 \left (-\frac {x^6}{210}-\frac {x^5}{60}+\frac {x^4}{20}+\frac {x^3}{2}+x^2\right ) \]