6.16.42 problem 38

Internal problem ID [2104]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS III. Exercises 7.7. Page 389
Problem number : 38
Date solved : Tuesday, September 30, 2025 at 05:24:04 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+5\right ) y^{\prime }-21 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 35

Order:=6; 
ode:=x^2*(x^2+1)*diff(diff(y(x),x),x)+x*(2*x^2+5)*diff(y(x),x)-21*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{3} \left (1-\frac {1}{2} x^{2}+\frac {15}{56} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-1316818944000-3456649728000 x^{2}-2880541440000 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{7}} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 46

ode=x^2*(1+x^2)*D[y[x],{x,2}]+x*(5+2*x^2)*D[y[x],x]-21*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{x^7}+\frac {21}{8 x^5}+\frac {35}{16 x^3}\right )+c_2 \left (\frac {15 x^7}{56}-\frac {x^5}{2}+x^3\right ) \]

✓ Sympy. Time used: 0.410 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**2 + 1)*Derivative(y(x), (x, 2)) + x*(2*x**2 + 5)*Derivative(y(x), x) - 21*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{3} + \frac {C_{1}}{x^{7}} + O\left (x^{6}\right ) \]