12.15.62 problem 63

Internal problem ID [2060]
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 : 63
Date solved : Monday, January 27, 2025 at 05:41:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (x^{2}+3 x +1\right ) y^{\prime \prime }+8 x^{2} \left (2 x +3\right ) y^{\prime }+\left (9 x^{2}+3 x +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 44

Order:=6; 
dsolve(4*x^2*(1+3*x+x^2)*diff(y(x),x$2)+8*x^2*(3+2*x)*diff(y(x),x)+(1+3*x+9*x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (-144 x^{5}+55 x^{4}-21 x^{3}+8 x^{2}-3 x +1\right ) \left (c_2 \ln \left (x \right )+c_1 \right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 72

AsymptoticDSolveValue[4*x^2*(1+3*x+x^2)*D[y[x],{x,2}]+8*x^2*(3+2*x)*D[y[x],x]+(1+3*x+9*x^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \sqrt {x} \left (-144 x^5+55 x^4-21 x^3+8 x^2-3 x+1\right )+c_2 \sqrt {x} \left (-144 x^5+55 x^4-21 x^3+8 x^2-3 x+1\right ) \log (x) \]