6.15.15 problem 11

Internal problem ID [2013]
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 II. Exercises 7.6. Page 374
Problem number : 11
Date solved : Tuesday, September 30, 2025 at 05:22:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} \left (1+x \right ) y^{\prime \prime }+3 x \left (-x^{2}+11 x +5\right ) y^{\prime }+\left (-7 x^{2}+16 x +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 54

Order:=8; 
ode:=9*x^2*(1+x)*diff(diff(y(x),x),x)+3*x*(-x^2+11*x+5)*diff(y(x),x)+(-7*x^2+16*x+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-x +\frac {7}{6} x^{2}-\frac {23}{18} x^{3}+\frac {11}{8} x^{4}-\frac {1577}{1080} x^{5}+\frac {3319}{2160} x^{6}-\frac {72853}{45360} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-\frac {1}{12} x^{2}+\frac {13}{108} x^{3}-\frac {131}{864} x^{4}+\frac {11449}{64800} x^{5}-\frac {76919}{388800} x^{6}+\frac {4118557}{19051200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2}{x^{{1}/{3}}} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 167

ode=9*x^2*(1+x)*D[y[x],{x,2}]+3*x*(5+11*x-x^2)*D[y[x],x]+(1+16*x-7*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to \frac {c_1 \left (-\frac {72853 x^7}{45360}+\frac {3319 x^6}{2160}-\frac {1577 x^5}{1080}+\frac {11 x^4}{8}-\frac {23 x^3}{18}+\frac {7 x^2}{6}-x+1\right )}{\sqrt [3]{x}}+c_2 \left (\frac {\frac {4118557 x^7}{19051200}-\frac {76919 x^6}{388800}+\frac {11449 x^5}{64800}-\frac {131 x^4}{864}+\frac {13 x^3}{108}-\frac {x^2}{12}}{\sqrt [3]{x}}+\frac {\left (-\frac {72853 x^7}{45360}+\frac {3319 x^6}{2160}-\frac {1577 x^5}{1080}+\frac {11 x^4}{8}-\frac {23 x^3}{18}+\frac {7 x^2}{6}-x+1\right ) \log (x)}{\sqrt [3]{x}}\right ) \]

✓ Sympy. Time used: 0.575 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x**2*(x + 1)*Derivative(y(x), (x, 2)) + 3*x*(-x**2 + 11*x + 5)*Derivative(y(x), x) + (-7*x**2 + 16*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = \frac {C_{1}}{\sqrt [3]{x}} + O\left (x^{8}\right ) \]