36.6.13 problem 13

Internal problem ID [6374]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number : 13
Date solved : Monday, January 27, 2025 at 01:58:24 PM
CAS classification : [_separable]

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

Using series method with expansion around

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

With initial conditions

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

✓ Solution by Maple

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

Order:=6; 
dsolve([diff(x(t),t)+sin(t)*x(t)=0,x(0) = 1],x(t),type='series',t=0);
 

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

✓ Solution by Mathematica

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

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

\[ x(t)\to \frac {t^4}{6}-\frac {t^2}{2}+1 \]