Internal problem ID [14783]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 4. Higher Order Equations. Exercises 4.8, page 203
Problem number: 7.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-y^{\prime }-2 y={\mathrm e}^{-x}} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 55
Order:=6; dsolve(diff(y(x),x$2)-diff(y(x),x)-2*y(x)=exp(-x),y(x),type='series',x=0);
\[ y \left (x \right ) = \left (1+x^{2}+\frac {1}{3} x^{3}+\frac {1}{4} x^{4}+\frac {1}{12} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {5}{24} x^{4}+\frac {11}{120} x^{5}\right ) D\left (y \right )\left (0\right )+\frac {x^{2}}{2}+\frac {x^{4}}{8}+\frac {x^{5}}{60}+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.02 (sec). Leaf size: 87
AsymptoticDSolveValue[y''[x]-y'[x]-2*y[x]==Exp[-x],y[x],{x,0,5}]
\[ y(x)\to \frac {x^5}{60}+\frac {x^4}{8}+\frac {x^2}{2}+c_1 \left (\frac {x^5}{12}+\frac {x^4}{4}+\frac {x^3}{3}+x^2+1\right )+c_2 \left (\frac {11 x^5}{120}+\frac {5 x^4}{24}+\frac {x^3}{2}+\frac {x^2}{2}+x\right ) \]