6.14.40 problem 42

Internal problem ID [1981]
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 : 42
Date solved : Tuesday, September 30, 2025 at 05:22:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 36

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

\[ y = \frac {c_2 \,x^{{4}/{3}} \left (1-\frac {3}{10} x^{2}+\frac {39}{320} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1-\frac {3}{2} x^{2}+\frac {15}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 50

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

\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {39 x^4}{320}-\frac {3 x^2}{10}+1\right )+\frac {c_2 \left (\frac {15 x^4}{32}-\frac {3 x^2}{2}+1\right )}{x} \]

✓ Sympy. Time used: 0.450 (sec). Leaf size: 15

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

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