12.14.41 problem 43

Internal problem ID [1982]
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 : 43
Date solved : Tuesday, March 04, 2025 at 01:47:28 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x \left (x^{2}+1\right ) y^{\prime \prime }+\left (7 x^{2}+4\right ) y^{\prime }+8 x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.731 (sec). Leaf size: 32

Order:=6; 
ode:=x*(x^2+1)*diff(diff(y(x),x),x)+(7*x^2+4)*diff(y(x),x)+8*x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \left (1-\frac {4}{5} x^{2}+\frac {24}{35} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (12-6 x^{2}+\frac {9}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 42

ode=x*(1+x^2)*D[y[x],{x,2}]+(4+7*x^2)*D[y[x],x]+8*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{x^3}+\frac {3 x}{8}-\frac {1}{2 x}\right )+c_2 \left (\frac {24 x^4}{35}-\frac {4 x^2}{5}+1\right ) \]

✓ Sympy. Time used: 0.988 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x**2 + 1)*Derivative(y(x), (x, 2)) + 8*x*y(x) + (7*x**2 + 4)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \left (- \frac {64 x^{5}}{1575} + \frac {64 x^{4}}{315} - \frac {32 x^{3}}{45} + \frac {8 x^{2}}{5} - 2 x + 1\right ) + O\left (x^{6}\right ) \]