Internal problem ID [5273]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+{\mathrm e}^{x} y=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 1, y^{\prime }\relax (0) = 0] \end {align*}
With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.003 (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);
\[ y \relax (x ) = 1-\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{40} x^{5}+\mathrm {O}\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 56
AsymptoticDSolveValue[{y''[x]+Exp[x]*y[x]==0,{}},y[x],{x,0,5}]
\[ y(x)\to c_2 \left (-\frac {x^5}{60}-\frac {x^4}{12}-\frac {x^3}{6}+x\right )+c_1 \left (\frac {x^5}{40}-\frac {x^3}{6}-\frac {x^2}{2}+1\right ) \]