64.15.5 problem 5

Internal problem ID [13582]
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 : 5
Date solved : Tuesday, January 28, 2025 at 05:53:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 33

Order:=6; 
dsolve(2*x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-1)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} \left (1-\frac {1}{2} x^{2}+\frac {1}{40} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} x \left (1-\frac {1}{14} x^{2}+\frac {1}{616} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 48

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

\[ y(x)\to c_1 x \left (\frac {x^4}{616}-\frac {x^2}{14}+1\right )+\frac {c_2 \left (\frac {x^4}{40}-\frac {x^2}{2}+1\right )}{\sqrt {x}} \]