12.15.59 problem 60

Internal problem ID [2057]
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 : 60
Date solved : Monday, January 27, 2025 at 05:41:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 29

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

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 46

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

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