6.15.17 problem 13

Internal problem ID [2015]
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 : 13
Date solved : Tuesday, September 30, 2025 at 05:22:34 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 36 x^{2} \left (1-2 x \right ) y^{\prime \prime }+24 x \left (1-9 x \right ) y^{\prime }+\left (1-70 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 48

Order:=6; 
ode:=36*x^2*(1-2*x)*diff(diff(y(x),x),x)+24*x*(1-9*x)*diff(y(x),x)+(1-70*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\frac {8}{3} x +\frac {56}{9} x^{2}+\frac {1120}{81} x^{3}+\frac {7280}{243} x^{4}+\frac {46592}{729} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {2}{3} x -2 x^{2}-\frac {1192}{243} x^{3}-\frac {8168}{729} x^{4}-\frac {270112}{10935} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) x^{{1}/{6}} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 132

ode=36*x^2*(1-2*x)*D[y[x],{x,2}]+24*x*(1-9*x)*D[y[x],x]+(1-70*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [6]{x} \left (\frac {46592 x^5}{729}+\frac {7280 x^4}{243}+\frac {1120 x^3}{81}+\frac {56 x^2}{9}+\frac {8 x}{3}+1\right )+c_2 \left (\sqrt [6]{x} \left (-\frac {270112 x^5}{10935}-\frac {8168 x^4}{729}-\frac {1192 x^3}{243}-2 x^2-\frac {2 x}{3}\right )+\sqrt [6]{x} \left (\frac {46592 x^5}{729}+\frac {7280 x^4}{243}+\frac {1120 x^3}{81}+\frac {56 x^2}{9}+\frac {8 x}{3}+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.478 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(36*x**2*(1 - 2*x)*Derivative(y(x), (x, 2)) + 24*x*(1 - 9*x)*Derivative(y(x), x) + (1 - 70*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \sqrt [6]{x} + O\left (x^{6}\right ) \]