12.14.43 problem 45

Internal problem ID [1984]
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 : 45
Date solved : Tuesday, March 04, 2025 at 01:47:30 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (8 x^{2}+3\right ) y^{\prime }-\left (-4 x^{2}+3\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 33

Order:=6; 
ode:=2*x^2*(x^2+1)*diff(diff(y(x),x),x)+x*(8*x^2+3)*diff(y(x),x)-(-4*x^2+3)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1-\frac {1}{4} x^{2}+\frac {5}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{3}/{2}}}+c_2 x \left (1-\frac {2}{3} x^{2}+\frac {20}{39} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 48

ode=2*x^2*(1+x^2)*D[y[x],{x,2}]+x*(3+8*x^2)*D[y[x],x]-(3-4*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 x \left (\frac {20 x^4}{39}-\frac {2 x^2}{3}+1\right )+\frac {c_2 \left (\frac {5 x^4}{32}-\frac {x^2}{4}+1\right )}{x^{3/2}} \]

✓ Sympy. Time used: 1.193 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*(x**2 + 1)*Derivative(y(x), (x, 2)) + x*(8*x**2 + 3)*Derivative(y(x), x) - (3 - 4*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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