75.23.5 problem 728

Internal problem ID [17202]
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 : Tuesday, January 28, 2025 at 09:56:59 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 14

Order:=6; 
dsolve([diff(y(x),x$2)-diff(y(x),x)*sin(x)=0,y(0) = 0, D(y)(0) = 1],y(x),type='series',x=0);
 

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

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 19

AsymptoticDSolveValue[{D[y[x],{x,2}]+Sin[x]*D[y[x],x]==0,{y[0]==0,Derivative[1][y][0] ==1}},y[x],{x,0,"6"-1}]
 

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