12.14.45 problem 47

Internal problem ID [1986]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 47
Date solved : Monday, January 27, 2025 at 05:39:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 6 x^{2} y^{\prime \prime }+x \left (6 x^{2}+1\right ) y^{\prime }+\left (9 x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 35

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

\[ y = c_1 \,x^{{1}/{3}} \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \sqrt {x}\, \left (1-\frac {6}{13} x^{2}+\frac {36}{325} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 52

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

\[ y(x)\to c_1 \sqrt {x} \left (\frac {36 x^4}{325}-\frac {6 x^2}{13}+1\right )+c_2 \sqrt [3]{x} \left (\frac {x^4}{8}-\frac {x^2}{2}+1\right ) \]