1.13.24 problem 25

Internal problem ID [423]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.1 (Introduction). Problems at page 206
Problem number : 25
Date solved : Tuesday, September 30, 2025 at 03:58:36 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }&=y^{\prime }+y \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 18

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

\[ y = x +\frac {1}{2} x^{2}+\frac {1}{3} x^{3}+\frac {1}{8} x^{4}+\frac {1}{24} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 33

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

\[ y(x)\to \frac {x^5}{24}+\frac {x^4}{8}+\frac {x^3}{3}+\frac {x^2}{2}+x \]

✓ Sympy. Time used: 0.238 (sec). Leaf size: 42

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{12} + \frac {x^{3}}{6} + \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {x^{3}}{8} + \frac {x^{2}}{3} + \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]