69.23.5 problem 728

Internal problem ID [18489]
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 : 728
Date solved : Thursday, October 02, 2025 at 03:14:13 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }-\sin \left (x \right ) y^{\prime }&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

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

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

\[ y = x +\frac {1}{6} x^{3}+\frac {1}{60} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

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

ode=D[y[x],{x,2}]+Sin[x]*D[y[x],x]==0; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {x^5}{30}-\frac {x^3}{6}+x \]

✓ Sympy. Time used: 1.066 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sin(x)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{3} \sin ^{3}{\left (x \right )}}{6} + \frac {x^{2} \sin ^{2}{\left (x \right )}}{2} + x \sin {\left (x \right )} + 1\right ) + C_{1} x + O\left (x^{6}\right ) \]