12.14.21 problem 21

Internal problem ID [1962]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 21
Date solved : Tuesday, March 04, 2025 at 01:47:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (3+x \right ) y^{\prime \prime }+x \left (1+5 x \right ) y^{\prime }+\left (1+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 47

Order:=6; 
ode:=2*x^2*(x+3)*diff(diff(y(x),x),x)+x*(1+5*x)*diff(y(x),x)+(1+x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{1}/{3}} \left (1-\frac {4}{9} x +\frac {14}{81} x^{2}-\frac {140}{2187} x^{3}+\frac {455}{19683} x^{4}-\frac {1456}{177147} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \sqrt {x}\, \left (1-\frac {3}{7} x +\frac {15}{91} x^{2}-\frac {15}{247} x^{3}+\frac {27}{1235} x^{4}-\frac {297}{38285} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 90

ode=2*x^2*(3+x)*D[y[x],{x,2}]+x*(1+5*x)*D[y[x],x]+(1+x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {297 x^5}{38285}+\frac {27 x^4}{1235}-\frac {15 x^3}{247}+\frac {15 x^2}{91}-\frac {3 x}{7}+1\right )+c_2 \sqrt [3]{x} \left (-\frac {1456 x^5}{177147}+\frac {455 x^4}{19683}-\frac {140 x^3}{2187}+\frac {14 x^2}{81}-\frac {4 x}{9}+1\right ) \]

✓ Sympy. Time used: 1.001 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*(x + 3)*Derivative(y(x), (x, 2)) + x*(5*x + 1)*Derivative(y(x), x) + (x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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