12.16.39 problem 35

Internal problem ID [2101]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS III. Exercises 7.7. Page 389
Problem number : 35
Date solved : Monday, January 27, 2025 at 05:42:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (11 x^{2}+5\right ) y^{\prime }+24 x^{2} y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 30

Order:=6; 
dsolve(x^2*(1+x^2)*diff(y(x),x$2)+x*(5+11*x^2)*diff(y(x),x)+24*x^2*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_1 \left (1-2 x^{2}+3 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-144+432 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{4}} \]

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 27

AsymptoticDSolveValue[x^2*(1+x^2)*D[y[x],{x,2}]+x*(5+11*x^2)*D[y[x],x]+24*x^2*y[x]==0,y[x],{x,0,"6"-1}]
 

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