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 : Sunday, March 30, 2025 at 01:38:01 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*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 48

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

\[ y = \frac {c_2 \,x^{{2}/{3}} \left (1+\frac {2}{15} x +\frac {11}{360} x^{2}+\frac {1}{162} x^{3}+\frac {29}{27216} x^{4}+\frac {551}{3470040} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1-\frac {4}{3} x -\frac {5}{18} x^{2}-\frac {5}{81} x^{3}-\frac {23}{1944} x^{4}-\frac {92}{47385} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{3}}} \]

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

ode=9*x^2*D[y[x],{x,2}]+9*(x-x^2)*D[y[x],x]+(x-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 1.062 (sec). Leaf size: 78

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x**2*Derivative(y(x), (x, 2)) + (x - 1)*y(x) + (-9*x**2 + 9*x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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