56.4.51 problem 48

Internal problem ID [8940]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 48
Date solved : Sunday, March 30, 2025 at 01:55:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 34

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

\[ y = c_1 x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (x +\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_2 +\left (1-x +\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.049 (sec). Leaf size: 20

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

\[ y(x)\to c_2 x+c_1 (-3 x+x \log (x)+1) \]

✓ Sympy. Time used: 0.911 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{2} x \left (\frac {x^{4}}{120} + \frac {x^{3}}{24} + \frac {x^{2}}{6} + \frac {x}{2} + 1\right ) + C_{1} + O\left (x^{6}\right ) \]