15.23.16 problem 20

Internal problem ID [3366]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 41, page 195
Problem number : 20
Date solved : Sunday, March 30, 2025 at 01:38:08 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (4+x \right ) x^{2} y^{\prime \prime }+x \left (x -1\right ) 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: 45

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

\[ y = c_1 \,x^{{1}/{4}} \left (1-\frac {1}{16} x +\frac {5}{512} x^{2}-\frac {15}{8192} x^{3}+\frac {195}{524288} x^{4}-\frac {663}{8388608} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 x \left (1-\frac {1}{7} x +\frac {2}{77} x^{2}-\frac {2}{385} x^{3}+\frac {8}{7315} x^{4}-\frac {8}{33649} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 86

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

\[ y(x)\to c_1 x \left (-\frac {8 x^5}{33649}+\frac {8 x^4}{7315}-\frac {2 x^3}{385}+\frac {2 x^2}{77}-\frac {x}{7}+1\right )+c_2 \sqrt [4]{x} \left (-\frac {663 x^5}{8388608}+\frac {195 x^4}{524288}-\frac {15 x^3}{8192}+\frac {5 x^2}{512}-\frac {x}{16}+1\right ) \]

✓ Sympy. Time used: 0.898 (sec). Leaf size: 15

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

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