51.4.13 problem 13

Internal problem ID [10370]
Book : First order enumerated odes
Section : section 4. First order odes solved using series method
Problem number : 13
Date solved : Tuesday, September 30, 2025 at 07:22:41 PM
CAS classification : [_linear]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.055 (sec). Leaf size: 28

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

\[ y = \frac {c_1 \left (1+\operatorname {O}\left (x^{6}\right )\right )}{x}+x \left (\frac {1}{2}+\frac {1}{12} x^{2}+\frac {1}{45} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 36

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

\[ y(x)\to \frac {\frac {x^6}{45}+\frac {x^4}{12}+\frac {x^2}{2}}{x}+\frac {c_1}{x} \]

✗ Sympy

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

ValueError : ODE x*Derivative(y(x), x) + y(x) - tan(x) does not match hint 1st_power_series