Internal problem ID [2402]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition,
2015
Section: Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.2. page
739
Problem number: Problem 15.
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 the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 49
Order:=6; dsolve(diff(y(x),x$2)-exp(x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{24} x^{5}\right ) y \relax (0)+\left (x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 63
AsymptoticDSolveValue[y''[x]-Exp[x]*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^5}{30}+\frac {x^4}{12}+\frac {x^3}{6}+x\right )+c_1 \left (\frac {x^5}{24}+\frac {x^4}{12}+\frac {x^3}{6}+\frac {x^2}{2}+1\right ) \]