74.16.25 problem 27 (a)

Internal problem ID [16531]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.8, page 203
Problem number : 27 (a)
Date solved : Tuesday, January 28, 2025 at 09:10:17 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 19

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

\[ y(x)\to \frac {x^3}{6}+\frac {x^2}{2}+1 \]