12.11.13 problem 24

Internal problem ID [1852]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number : 24
Date solved : Tuesday, March 04, 2025 at 01:45:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x^{2} \left (1+3 x \right ) y^{\prime \prime }+x \left (x^{2}+12 x +2\right ) y^{\prime }+2 x \left (3+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 44

Order:=6; 
ode:=x^2*(1+3*x)*diff(diff(y(x),x),x)+x*(x^2+12*x+2)*diff(y(x),x)+2*x*(x+3)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \left (1-3 x +\frac {26}{3} x^{2}-\frac {101}{4} x^{3}+\frac {4441}{60} x^{4}-\frac {26141}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (1-6 x +\frac {35}{2} x^{2}-\frac {101}{2} x^{3}+\frac {1177}{8} x^{4}-\frac {17251}{40} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.051 (sec). Leaf size: 60

ode=x^2*(1+3*x)*D[y[x],{x,2}]+x*(2+12*x+x^2)*D[y[x],x]+2*x*(3+x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {571 x^3}{8}-\frac {49 x^2}{2}+\frac {17 x}{2}+\frac {1}{x}-3\right )+c_2 \left (\frac {4441 x^4}{60}-\frac {101 x^3}{4}+\frac {26 x^2}{3}-3 x+1\right ) \]

✓ Sympy. Time used: 1.177 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(3*x + 1)*Derivative(y(x), (x, 2)) + 2*x*(x + 3)*y(x) + x*(x**2 + 12*x + 2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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