42.1.19 problem 3.48 (a)

Internal problem ID [6841]
Book : Advanced Mathematical Methods for Scientists and Engineers, Bender and Orszag. Springer October 29, 1999
Section : Chapter 3. APPROXIMATE SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS. page 136
Problem number : 3.48 (a)
Date solved : Sunday, March 30, 2025 at 11:24:13 AM
CAS classification : [_linear]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 35

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

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) y \left (0\right )+x -\frac {x^{3}}{2}+\frac {13 x^{5}}{120}+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 38

ode=D[y[x],x]+x*y[x]==Cos[x]; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {13 x^5}{120}-\frac {x^3}{2}+c_1 \left (\frac {x^4}{8}-\frac {x^2}{2}+1\right )+x \]

✓ Sympy. Time used: 0.853 (sec). Leaf size: 34

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

\[ y{\left (x \right )} = x - \frac {x^{3}}{2} + \frac {13 x^{5}}{120} + C_{1} - \frac {C_{1} x^{2}}{2} + \frac {C_{1} x^{4}}{8} + O\left (x^{6}\right ) \]