36.6.17 problem 19

Internal problem ID [6378]
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 : 19
Date solved : Monday, January 27, 2025 at 01:58:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-{\mathrm e}^{2 x} y^{\prime }+y \cos \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 )&=-1\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 20

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

\[ y = -1+x +x^{2}+\frac {1}{2} x^{3}+\frac {1}{2} x^{4}+\frac {31}{60} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 30

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

\[ y(x)\to \frac {31 x^5}{60}+\frac {x^4}{2}+\frac {x^3}{2}+x^2+x-1 \]