12.15.33 problem 29

Internal problem ID [2031]
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 II. Exercises 7.6. Page 374
Problem number : 29
Date solved : Monday, January 27, 2025 at 05:40:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-3 x \left (-x^{2}+1\right ) y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 36

Order:=6; 
dsolve(x^2*(1+x^2)*diff(y(x),x$2)-3*x*(1-x^2)*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
 

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 61

AsymptoticDSolveValue[x^2*(1+x^2)*D[y[x],{x,2}]-3*x*(1-x^2)*D[y[x],x]+4*y[x]==0,y[x],{x,0,"6"-1}]
 

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