12.14.19 problem 19

Internal problem ID [1960]
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 : 19
Date solved : Monday, January 27, 2025 at 05:39:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 48

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

\[ y = \frac {c_2 \,x^{{2}/{3}} \left (1+\frac {1}{5} x +\frac {1}{80} x^{2}+\frac {1}{2640} x^{3}+\frac {1}{147840} x^{4}+\frac {1}{12566400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+x +\frac {1}{8} x^{2}+\frac {1}{168} x^{3}+\frac {1}{6720} x^{4}+\frac {1}{436800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{3}}} \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 86

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

\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {x^5}{12566400}+\frac {x^4}{147840}+\frac {x^3}{2640}+\frac {x^2}{80}+\frac {x}{5}+1\right )+\frac {c_2 \left (\frac {x^5}{436800}+\frac {x^4}{6720}+\frac {x^3}{168}+\frac {x^2}{8}+x+1\right )}{\sqrt [3]{x}} \]