12.13.38 problem 38

Internal problem ID [1929]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN ORDINARY POINT II. Exercises 7.3. Page 338
Problem number : 38
Date solved : Tuesday, March 04, 2025 at 01:46:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.000 (sec). Leaf size: 20

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

\[ y = -2+3 x +\frac {4}{3} x^{2}-x^{3}-\frac {19}{54} x^{4}+\frac {13}{60} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 34

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

\[ y(x)\to \frac {13 x^5}{60}-\frac {19 x^4}{54}-x^3+\frac {4 x^2}{3}+3 x-2 \]

✓ Sympy. Time used: 0.834 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = - \frac {x^{5} r{\left (3 \right )}}{6} + C_{2} \left (\frac {19 x^{4}}{108} - \frac {2 x^{2}}{3} + 1\right ) + C_{1} x \left (\frac {x^{4}}{60} + 1\right ) + O\left (x^{6}\right ) \]