12.13.15 problem 15

Internal problem ID [1906]
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 : 15
Date solved : Tuesday, March 04, 2025 at 01:45:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.000 (sec). Leaf size: 14

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

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 17

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

\[ y(x)\to \frac {3 x^4}{2}-2 x^2+1 \]

✓ Sympy. Time used: 0.781 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = - \frac {13 x^{5} r{\left (3 \right )}}{20} + C_{2} \left (\frac {3 x^{4}}{2} - 2 x^{2} + 1\right ) + C_{1} x \left (1 - \frac {x^{4}}{10}\right ) + O\left (x^{6}\right ) \]