56.4.69 problem 66

Internal problem ID [8958]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 66
Date solved : Sunday, March 30, 2025 at 01:56:27 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 54

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

\[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{30} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {1}{120} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 63

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

\[ y(x)\to c_2 \left (\frac {x^5}{120}-\frac {x^4}{12}+\frac {x^3}{6}+x\right )+c_1 \left (-\frac {x^5}{30}+\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.780 (sec). Leaf size: 39

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

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