12.16.33 problem 29

Internal problem ID [2095]
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 : 29
Date solved : Saturday, March 29, 2025 at 11:48:22 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (x^{2}+1\right ) y^{\prime }-\left (-3 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: 46

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

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 51

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

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

✓ Sympy. Time used: 0.894 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = C_{1} x \left (\frac {x^{4}}{8} - \frac {x^{2}}{2} + 1\right ) + O\left (x^{6}\right ) \]