12.15.39 problem 35

Internal problem ID [2037]
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 : 35
Date solved : Monday, January 27, 2025 at 05:40:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} \left (x^{2}+3\right ) y^{\prime \prime }+x \left (11 x^{2}+3\right ) y^{\prime }+\left (5 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: 36

Order:=6; 
dsolve(3*x^2*(3+x^2)*diff(y(x),x$2)+x*(3+11*x^2)*diff(y(x),x)+(1+5*x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = x^{{1}/{3}} \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {2}{9} x^{2}+\frac {5}{81} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{18} x^{2}+\frac {1}{54} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 42

AsymptoticDSolveValue[3*x^2*(3+x^2)*D[y[x],x]+x*(3+11*x^2)*D[y[x],x]+(1+5*x^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to \frac {c_1 \left (-\frac {1139 x^5}{405}+\frac {53 x^4}{81}+\frac {7 x^3}{9}-\frac {11 x^2}{9}+x+1\right )}{\sqrt [3]{x}} \]