6.15.10 problem 6

Internal problem ID [2008]
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 : 6
Date solved : Tuesday, September 30, 2025 at 05:22:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} y^{\prime \prime }+3 x \left (-2 x^{2}+3 x +5\right ) y^{\prime }+\left (-14 x^{2}+12 x +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.042 (sec). Leaf size: 56

Order:=8; 
ode:=9*x^2*diff(diff(y(x),x),x)+3*x*(-2*x^2+3*x+5)*diff(y(x),x)+(-14*x^2+12*x+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-x +\frac {5}{6} x^{2}-\frac {1}{2} x^{3}+\frac {19}{72} x^{4}-\frac {43}{360} x^{5}+\frac {319}{6480} x^{6}-\frac {167}{9072} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (x -\frac {11}{12} x^{2}+\frac {25}{36} x^{3}-\frac {113}{288} x^{4}+\frac {4211}{21600} x^{5}-\frac {32773}{388800} x^{6}+\frac {126647}{3810240} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2}{x^{{1}/{3}}} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 168

ode=9*x^2*D[y[x],{x,2}]+3*x*(5+3*x-2*x^2)*D[y[x],x]+(1+12*x-14*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to \frac {c_1 \left (-\frac {167 x^7}{9072}+\frac {319 x^6}{6480}-\frac {43 x^5}{360}+\frac {19 x^4}{72}-\frac {x^3}{2}+\frac {5 x^2}{6}-x+1\right )}{\sqrt [3]{x}}+c_2 \left (\frac {\frac {126647 x^7}{3810240}-\frac {32773 x^6}{388800}+\frac {4211 x^5}{21600}-\frac {113 x^4}{288}+\frac {25 x^3}{36}-\frac {11 x^2}{12}+x}{\sqrt [3]{x}}+\frac {\left (-\frac {167 x^7}{9072}+\frac {319 x^6}{6480}-\frac {43 x^5}{360}+\frac {19 x^4}{72}-\frac {x^3}{2}+\frac {5 x^2}{6}-x+1\right ) \log (x)}{\sqrt [3]{x}}\right ) \]

✓ Sympy. Time used: 0.444 (sec). Leaf size: 54

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x**2*Derivative(y(x), (x, 2)) + 3*x*(-2*x**2 + 3*x + 5)*Derivative(y(x), x) + (-14*x**2 + 12*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = \frac {C_{1} \left (- \frac {167 x^{7}}{9072} + \frac {319 x^{6}}{6480} - \frac {43 x^{5}}{360} + \frac {19 x^{4}}{72} - \frac {x^{3}}{2} + \frac {5 x^{2}}{6} - x + 1\right )}{\sqrt [3]{x}} + O\left (x^{8}\right ) \]