12.15.55 problem 51

Internal problem ID [2053]
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 : 51
Date solved : Monday, January 27, 2025 at 05:41:10 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x \left (x^{2}+1\right ) y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }-8 y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 32

Order:=6; 
dsolve(x*(1+x^2)*diff(y(x),x$2)+(1-x^2)*diff(y(x),x)-8*x*y(x)=0,y(x),type='series',x=0);
 

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 48

AsymptoticDSolveValue[x*(1+x^2)*D[y[x],{x,2}]+(1-x^2)*D[y[x],x]-8*x*y[x]==0,y[x],{x,0,"6"-1}]
 

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