12.11.2 problem 12

Internal problem ID [1841]
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 : 12
Date solved : Tuesday, March 04, 2025 at 01:44:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 47

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

\[ y = \left (1+x^{2}-\frac {1}{3} x^{4}-\frac {3}{10} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}-\frac {1}{4} x^{4}-\frac {4}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 52

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

\[ y(x)\to c_2 \left (-\frac {4 x^5}{15}-\frac {x^4}{4}+\frac {x^3}{3}+x\right )+c_1 \left (-\frac {3 x^5}{10}-\frac {x^4}{3}+x^2+1\right ) \]

✓ Sympy. Time used: 0.857 (sec). Leaf size: 32

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

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