52.2.23 problem 23

Internal problem ID [8274]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS ABOUT SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number : 23
Date solved : Wednesday, March 05, 2025 at 05:33:49 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} y^{\prime \prime }+9 x^{2} y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 55

Order:=8; 
ode:=9*x^2*diff(diff(y(x),x),x)+9*x^2*diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_{1} x^{{1}/{3}} \left (1-\frac {1}{2} x +\frac {1}{5} x^{2}-\frac {7}{120} x^{3}+\frac {7}{528} x^{4}-\frac {13}{5280} x^{5}+\frac {13}{33660} x^{6}-\frac {247}{4712400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{{2}/{3}} \left (1-\frac {1}{2} x +\frac {5}{28} x^{2}-\frac {1}{21} x^{3}+\frac {11}{1092} x^{4}-\frac {11}{6240} x^{5}+\frac {187}{711360} x^{6}-\frac {17}{497952} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 118

ode=9*x^2*D[y[x],{x,2}]+9*x^2*D[y[x],x]+2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \sqrt [3]{x} \left (-\frac {247 x^7}{4712400}+\frac {13 x^6}{33660}-\frac {13 x^5}{5280}+\frac {7 x^4}{528}-\frac {7 x^3}{120}+\frac {x^2}{5}-\frac {x}{2}+1\right )+c_1 x^{2/3} \left (-\frac {17 x^7}{497952}+\frac {187 x^6}{711360}-\frac {11 x^5}{6240}+\frac {11 x^4}{1092}-\frac {x^3}{21}+\frac {5 x^2}{28}-\frac {x}{2}+1\right ) \]

✓ Sympy. Time used: 1.131 (sec). Leaf size: 94

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x**2*Derivative(y(x), x) + 9*x**2*Derivative(y(x), (x, 2)) + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {2}{3}} \left (\frac {187 x^{6}}{711360} - \frac {11 x^{5}}{6240} + \frac {11 x^{4}}{1092} - \frac {x^{3}}{21} + \frac {5 x^{2}}{28} - \frac {x}{2} + 1\right ) + C_{1} \sqrt [3]{x} \left (\frac {13 x^{6}}{33660} - \frac {13 x^{5}}{5280} + \frac {7 x^{4}}{528} - \frac {7 x^{3}}{120} + \frac {x^{2}}{5} - \frac {x}{2} + 1\right ) + O\left (x^{8}\right ) \]