64.15.17 problem 17

Internal problem ID [13523]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius). Exercises page 251
Problem number : 17
Date solved : Monday, March 31, 2025 at 08:00:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 44

Order:=6; 
ode:=(2*x^2-x)*diff(diff(y(x),x),x)+(2*x-2)*diff(y(x),x)+(-2*x^2+3*x-2)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (1-2 x +\frac {7}{2} x^{2}-\frac {4}{3} x^{3}+\frac {13}{24} x^{4}-\frac {7}{60} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 60

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

\[ y(x)\to c_1 \left (\frac {7 x^3}{8}-\frac {7 x^2}{3}+\frac {11 x}{2}+\frac {1}{x}-4\right )+c_2 \left (\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}-x+1\right ) \]

✓ Sympy. Time used: 1.029 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((2*x - 2)*Derivative(y(x), x) + (2*x**2 - x)*Derivative(y(x), (x, 2)) + (-2*x**2 + 3*x - 2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {C_{2}}{x} + C_{1} + O\left (x^{6}\right ) \]