64.15.15 problem 15

Internal problem ID [13592]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius). Exercises page 251
Problem number : 15
Date solved : Tuesday, January 28, 2025 at 05:53:30 AM
CAS classification : [_Lienard]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 32

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

\[ y = c_{1} x^{3} \left (1+\frac {1}{5} x^{2}+\frac {1}{35} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (12+6 x^{2}+\frac {3}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 44

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

\[ y(x)\to c_1 \left (\frac {x^4}{8}+\frac {x^2}{2}+1\right )+c_2 \left (\frac {x^7}{35}+\frac {x^5}{5}+x^3\right ) \]