9.8.24 problem problem 24

Internal problem ID [1089]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Chapter 11 Power series methods. Section 11.2 Power series solutions. Page 624
Problem number : problem 24
Date solved : Monday, January 27, 2025 at 04:33:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 44

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

\[ y = \left (1+\frac {1}{3} x^{3}+\frac {1}{5} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}+\frac {1}{6} x^{4}+\frac {1}{5} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 49

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

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