6.15.37 problem 33

Internal problem ID [2035]
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 II. Exercises 7.6. Page 374
Problem number : 33
Date solved : Tuesday, September 30, 2025 at 05:22:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-4 x^{2}+1\right ) y^{\prime }+\left (2 x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 34

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

\[ y = x \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {3}{2} x^{2}+\frac {15}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {13}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 65

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

\[ y(x)\to c_1 x \left (\frac {15 x^4}{8}-\frac {3 x^2}{2}+1\right )+c_2 \left (x \left (\frac {x^2}{4}-\frac {13 x^4}{32}\right )+x \left (\frac {15 x^4}{8}-\frac {3 x^2}{2}+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.434 (sec). Leaf size: 8

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

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