6.15.5 problem 1

Internal problem ID [2003]
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 : 1
Date solved : Tuesday, September 30, 2025 at 05:22:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 54

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

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-x +\frac {3}{4} x^{2}-\frac {13}{36} x^{3}+\frac {79}{576} x^{4}-\frac {67}{1600} x^{5}+\frac {5593}{518400} x^{6}-\frac {60859}{25401600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (x -x^{2}+\frac {65}{108} x^{3}-\frac {895}{3456} x^{4}+\frac {12547}{144000} x^{5}-\frac {41729}{1728000} x^{6}+\frac {10121677}{1778112000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \right ) x \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 154

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

\[ y(x)\to c_1 x \left (-\frac {60859 x^7}{25401600}+\frac {5593 x^6}{518400}-\frac {67 x^5}{1600}+\frac {79 x^4}{576}-\frac {13 x^3}{36}+\frac {3 x^2}{4}-x+1\right )+c_2 \left (x \left (\frac {10121677 x^7}{1778112000}-\frac {41729 x^6}{1728000}+\frac {12547 x^5}{144000}-\frac {895 x^4}{3456}+\frac {65 x^3}{108}-x^2+x\right )+x \left (-\frac {60859 x^7}{25401600}+\frac {5593 x^6}{518400}-\frac {67 x^5}{1600}+\frac {79 x^4}{576}-\frac {13 x^3}{36}+\frac {3 x^2}{4}-x+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.339 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = C_{1} x \left (\frac {5593 x^{6}}{518400} - \frac {67 x^{5}}{1600} + \frac {79 x^{4}}{576} - \frac {13 x^{3}}{36} + \frac {3 x^{2}}{4} - x + 1\right ) + O\left (x^{8}\right ) \]