49.15.8 problem 4

Internal problem ID [7694]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 3. Linear equations with variable coefficients. Page 130
Problem number : 4
Date solved : Sunday, March 30, 2025 at 12:19:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+{\mathrm e}^{x} 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: 0.006 (sec). Leaf size: 16

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

\[ y = 1-\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{40} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 26

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

\[ y(x)\to \frac {x^5}{40}-\frac {x^3}{6}-\frac {x^2}{2}+1 \]

✓ Sympy. Time used: 0.704 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)*exp(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)
 

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