12.16.36 problem 32

Internal problem ID [2098]
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 III. Exercises 7.7. Page 389
Problem number : 32
Date solved : Monday, January 27, 2025 at 05:42:17 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} y^{\prime \prime }-3 x \left (2 x^{2}+11\right ) y^{\prime }+\left (10 x^{2}+13\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: 46

Order:=6; 
dsolve(9*x^2*diff(y(x),x$2)-3*x*(11+2*x^2)*diff(y(x),x)+(13+10*x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (x^{4} \left (1+\frac {4}{27} x^{2}+\frac {7}{486} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_1 +c_2 \left (\ln \left (x \right ) \left (-\frac {32}{9} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-32 x^{2}-\frac {8}{3} x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) x^{{1}/{3}} \]

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 62

AsymptoticDSolveValue[9*x^2*D[y[x],{x,2}]-3*x*(11+2*x^2)*D[y[x],x]+(13+10*x^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {7 x^{25/3}}{486}+\frac {4 x^{19/3}}{27}+x^{13/3}\right )+c_1 \left (\frac {2}{81} x^{13/3} \log (x)+\frac {1}{81} \left (x^2+9\right )^2 \sqrt [3]{x}\right ) \]