12.15.58 problem 59

Internal problem ID [2056]
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 : 59
Date solved : Monday, January 27, 2025 at 05:41:13 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} \left (x +3\right ) y^{\prime \prime }+3 x \left (3+7 x \right ) y^{\prime }+\left (3+4 x \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(9*x^2*(3+x)*diff(y(x),x$2)+3*x*(3+7*x)*diff(y(x),x)+(3+4*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = x^{{1}/{3}} \left (1-\frac {1}{3} x +\frac {1}{9} x^{2}-\frac {1}{27} x^{3}+\frac {1}{81} x^{4}-\frac {1}{243} x^{5}\right ) \left (c_2 \ln \left (x \right )+c_1 \right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 92

AsymptoticDSolveValue[9*x^2*(3+x)*D[y[x],{x,2}]+3*x*(3+7*x)*D[y[x],x]+(3+4*x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {x^5}{243}+\frac {x^4}{81}-\frac {x^3}{27}+\frac {x^2}{9}-\frac {x}{3}+1\right )+c_2 \sqrt [3]{x} \left (-\frac {x^5}{243}+\frac {x^4}{81}-\frac {x^3}{27}+\frac {x^2}{9}-\frac {x}{3}+1\right ) \log (x) \]