15.23.12 problem 16

Internal problem ID [3362]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 41, page 195
Problem number : 16
Date solved : Monday, January 27, 2025 at 07:34:55 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 48

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

\[ y \left (x \right ) = \frac {c_{2} x^{{2}/{3}} \left (1+\frac {2}{15} x +\frac {11}{360} x^{2}+\frac {1}{162} x^{3}+\frac {29}{27216} x^{4}+\frac {551}{3470040} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {4}{3} x -\frac {5}{18} x^{2}-\frac {5}{81} x^{3}-\frac {23}{1944} x^{4}-\frac {92}{47385} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{3}}} \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 90

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

\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {551 x^5}{3470040}+\frac {29 x^4}{27216}+\frac {x^3}{162}+\frac {11 x^2}{360}+\frac {2 x}{15}+1\right )+\frac {c_2 \left (-\frac {92 x^5}{47385}-\frac {23 x^4}{1944}-\frac {5 x^3}{81}-\frac {5 x^2}{18}-\frac {4 x}{3}+1\right )}{\sqrt [3]{x}} \]