15.24.14 problem 14

Internal problem ID [3386]
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 : 14
Date solved : Tuesday, March 04, 2025 at 04:37:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (x -4\right ) y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 60

Order:=6; 
ode:=x^2*(1+x)*diff(diff(y(x),x),x)+x*(x-4)*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y \left (x \right ) = x \left (c_{1} x^{3} \left (1-4 x +10 x^{2}-20 x^{3}+35 x^{4}-56 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\left (-36\right ) x^{3}+144 x^{4}-360 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+6 x +12 x^{2}-240 x^{3}+852 x^{4}-2022 x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 70

ode=x^2*(x+1)*D[y[x],{x,2}]+x*(x-4)*D[y[x],x]+4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (3 (4 x-1) x^4 \log (x)+\frac {1}{2} \left (62 x^4-20 x^3+2 x^2+x+2\right ) x\right )+c_2 \left (35 x^8-20 x^7+10 x^6-4 x^5+x^4\right ) \]

✓ Sympy. Time used: 1.064 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x + 1)*Derivative(y(x), (x, 2)) + x*(x - 4)*Derivative(y(x), x) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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