15.24.8 problem 8

Internal problem ID [3380]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 42, page 206
Problem number : 8
Date solved : Tuesday, March 04, 2025 at 04:37:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 42

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

\[ y \left (x \right ) = c_{1} x^{2} \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (-2+2 x -\frac {2}{3} x^{3}+\frac {5}{12} x^{4}-\frac {3}{20} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 59

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

\[ y(x)\to c_1 \left (-\frac {5 x^4}{24}+\frac {x^3}{3}-x+1\right )+c_2 \left (\frac {x^6}{24}-\frac {x^5}{6}+\frac {x^4}{2}-x^3+x^2\right ) \]

✓ Sympy. Time used: 0.927 (sec). Leaf size: 48

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

\[ y{\left (x \right )} = C_{2} \left (\frac {3 x^{5}}{40} - \frac {5 x^{4}}{24} + \frac {x^{3}}{3} - x + 1\right ) + C_{1} x^{2} \left (- \frac {x^{3}}{6} + \frac {x^{2}}{2} - x + 1\right ) + O\left (x^{6}\right ) \]