69.23.3 problem 726

Internal problem ID [18487]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 18.1 Integration of differential equation in series. Power series. Exercises page 171
Problem number : 726
Date solved : Thursday, October 02, 2025 at 03:14:12 PM
CAS classification : [_separable]

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

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 14

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

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 19

ode=D[y[x],x]==Sin[x]*y[x]; 
ic={y[0]==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {x^4}{12}+\frac {x^2}{2}+1 \]

✓ Sympy. Time used: 1.041 (sec). Leaf size: 17

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

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