30.14.14 problem 14

Internal problem ID [7663]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number : 14
Date solved : Tuesday, September 30, 2025 at 04:55:38 PM
CAS classification : [_separable]

\begin{align*} y^{\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 \\ \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

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

\[ y = 1+x +x^{2}+\frac {5}{6} x^{3}+\frac {5}{8} x^{4}+\frac {13}{30} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 30

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

\[ y(x)\to \frac {13 x^5}{30}+\frac {5 x^4}{8}+\frac {5 x^3}{6}+x^2+x+1 \]

✓ Sympy. Time used: 0.161 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = 1 + x + x^{2} + \frac {5 x^{3}}{6} + \frac {5 x^{4}}{8} + \frac {13 x^{5}}{30} + O\left (x^{6}\right ) \]