12.11.14 problem 25

Internal problem ID [1853]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number : 25
Date solved : Monday, January 27, 2025 at 05:37:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+4\right ) y^{\prime }+2 \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: 0.008 (sec). Leaf size: 30

Order:=6; 
dsolve(x^2*(1+2*x^2)*diff(y(x),x$2)+x*(4+2*x^2)*diff(y(x),x)+2*(1-x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_1 x \left (1+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-3 x^{2}-\frac {1}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.017 (sec). Leaf size: 25

AsymptoticDSolveValue[x^2*(1+2*x^2)*D[y[x],{x,2}]+x*(4+2*x^2)*D[y[x],x]+2*(1-x^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (-\frac {x^2}{2}+\frac {1}{x^2}-3\right )+\frac {c_2}{x} \]