73.23.29 problem 33.11 (c)

Internal problem ID [15604]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 33. Power series solutions I: Basic computational methods. Additional Exercises. page 641
Problem number : 33.11 (c)
Date solved : Monday, March 31, 2025 at 01:42:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+{\mathrm e}^{2 x} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 34

Order:=4; 
ode:=diff(diff(y(x),x),x)+exp(2*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 35

ode=D[y[x],{x,2}]+Exp[2*x]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,3}]
 

\[ y(x)\to c_2 \left (x-\frac {x^3}{6}\right )+c_1 \left (-\frac {x^3}{3}-\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.414 (sec). Leaf size: 22

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

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