49.15.8 problem 4

Internal problem ID [7694]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 3. Linear equations with variable coefficients. Page 130
Problem number : 4
Date solved : Monday, January 27, 2025 at 03:09:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+{\mathrm e}^{x} y&=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 )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 16

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

\[ y = 1-\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{40} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 56

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

\[ y(x)\to c_2 \left (-\frac {x^5}{60}-\frac {x^4}{12}-\frac {x^3}{6}+x\right )+c_1 \left (\frac {x^5}{40}-\frac {x^3}{6}-\frac {x^2}{2}+1\right ) \]