12.11.5 problem 15

Internal problem ID [1844]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number : 15
Date solved : Tuesday, March 04, 2025 at 01:44:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 37

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

\[ y = \left (x^{4}-2 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {4}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 36

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

\[ y(x)\to c_2 \left (\frac {4 x^5}{15}-\frac {x^3}{3}+x\right )+c_1 \left (x^4-2 x^2+1\right ) \]

✓ Sympy. Time used: 0.836 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = C_{2} \left (x^{4} - 2 x^{2} + 1\right ) + C_{1} x \left (1 - \frac {x^{2}}{3}\right ) + O\left (x^{6}\right ) \]