50.17.21 problem 4

Internal problem ID [8092]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.2. Series Solutions of First-Order Differential Equations Page 162
Problem number : 4
Date solved : Wednesday, March 05, 2025 at 05:29:43 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=1+y \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 76

Order:=8; 
ode:=diff(y(x),x) = 1+y(x); 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}\right ) y \left (0\right )+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 95

ode=D[y[x],x]==1+y[x]; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to \frac {x^7}{5040}+\frac {x^6}{720}+\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+c_1 \left (\frac {x^7}{5040}+\frac {x^6}{720}+\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right )+x \]

✓ Sympy. Time used: 0.705 (sec). Leaf size: 63

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x) + Derivative(y(x), x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=0,n=8)
 

\[ y{\left (x \right )} = x \left (C_{1} + 1\right ) + \frac {x^{2} \left (C_{1} + 1\right )}{2} + \frac {x^{3} \left (C_{1} + 1\right )}{6} + \frac {x^{4} \left (C_{1} + 1\right )}{24} + \frac {x^{5} \left (C_{1} + 1\right )}{120} + \frac {x^{6} \left (C_{1} + 1\right )}{720} + \frac {x^{7} \left (C_{1} + 1\right )}{5040} + C_{1} + O\left (x^{8}\right ) \]