62.24.1 problem Ex 1

Internal problem ID [12929]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VII, Linear differential equations with constant coefficients. Article 47. Particular integral. Page 100
Problem number : Ex 1
Date solved : Tuesday, January 28, 2025 at 04:44:14 AM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime }-2 y^{\prime }&={\mathrm e}^{-x} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 27

dsolve(diff(y(x),x$3)-diff(y(x),x$2)-2*diff(y(x),x)=exp(-x),y(x), singsol=all)
 

\[ y = \frac {\left (2 x -6 c_{2} +2\right ) {\mathrm e}^{-x}}{6}+\frac {{\mathrm e}^{2 x} c_{1}}{2}+c_3 \]

✓ Solution by Mathematica

Time used: 5.637 (sec). Leaf size: 125

DSolve[D[y[x],{x,3}]-D[y[x],{x,2}]-2*D[y[x],x]==Exp[-x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \int _1^x\frac {1}{9} e^{-K[1]} \left (9 c_1+9 e^{3 K[1]} c_2-3 K[1]-1\right )dK[1]+c_3 \\ y(x)\to \frac {1}{9} e^{-x} (3 x+4-9 c_1)+\frac {-\frac {7}{9}+c_1}{e}+c_3 \\ y(x)\to \frac {1}{18} \left (e^{-x} (6 x+8)+9 c_2 e^{2 x}-\frac {14}{e}-9 e^2 c_2+18 c_3\right ) \\ \end{align*}