51.4.2 problem 2

Internal problem ID [10359]
Book : First order enumerated odes
Section : section 4. First order odes solved using series method
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 07:22:35 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }+y&=\sin \left (x \right ) \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 47

Order:=6; 
ode:=diff(y(x),x)+y(x) = sin(x); 
dsolve(ode,y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 54

ode=D[y[x],x]+y[x]==Sin[x]; 
AsymptoticDSolveValue[ode,y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.181 (sec). Leaf size: 41

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

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