9.8.27 problem problem 27

Internal problem ID [1092]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Chapter 11 Power series methods. Section 11.2 Power series solutions. Page 624
Problem number : problem 27
Date solved : Saturday, March 29, 2025 at 10:38:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.067 (sec). Leaf size: 20

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

\[ y = 1-x -\frac {1}{2} x^{2}+\frac {1}{3} x^{3}-\frac {1}{24} x^{4}+\frac {1}{30} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

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

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

\[ y(x)\to \frac {x^5}{30}-\frac {x^4}{24}+\frac {x^3}{3}-\frac {x^2}{2}-x+1 \]

✓ Sympy. Time used: 0.776 (sec). Leaf size: 37

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

\[ y{\left (x \right )} = - \frac {x^{5} r{\left (3 \right )}}{5} + C_{2} \left (- \frac {x^{4}}{24} - \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (1 - \frac {x^{4}}{10}\right ) + O\left (x^{6}\right ) \]