56.4.14 problem 14

Internal problem ID [8903]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 14
Date solved : Sunday, March 30, 2025 at 01:52:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x -2\right ) y^{\prime \prime }+\frac {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)+1/x*diff(y(x),x)+(1+x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{3}/{2}} \left (1+\frac {3}{20} x +\frac {25}{224} x^{2}+\frac {1361}{17280} x^{3}+\frac {80753}{2365440} x^{4}+\frac {616517}{38707200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+\frac {1}{2} x^{2}+\frac {2}{9} x^{3}+\frac {11}{120} x^{4}+\frac {82}{1575} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 80

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

\[ y(x)\to c_2 \left (\frac {82 x^5}{1575}+\frac {11 x^4}{120}+\frac {2 x^3}{9}+\frac {x^2}{2}+1\right )+c_1 \left (\frac {616517 x^5}{38707200}+\frac {80753 x^4}{2365440}+\frac {1361 x^3}{17280}+\frac {25 x^2}{224}+\frac {3 x}{20}+1\right ) x^{3/2} \]

✓ Sympy. Time used: 1.057 (sec). Leaf size: 14

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

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