12.15.63 problem 64

Internal problem ID [2061]
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 : 64
Date solved : Monday, January 27, 2025 at 05:41:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-x \right )^{2} y^{\prime \prime }-x \left (-3 x^{2}+2 x +1\right ) y^{\prime }+\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: 0.007 (sec). Leaf size: 42

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

\[ y = \left (6 x^{5}+5 x^{4}+4 x^{3}+3 x^{2}+2 x +1\right ) x \left (c_2 \ln \left (x \right )+c_1 \right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 64

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

\[ y(x)\to c_1 x \left (6 x^5+5 x^4+4 x^3+3 x^2+2 x+1\right )+c_2 x \left (6 x^5+5 x^4+4 x^3+3 x^2+2 x+1\right ) \log (x) \]