7.11.38 problem 39

Internal problem ID [359]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 2. Linear Equations of Higher Order. Section 2.5 (Nonhomogeneous equations and undetermined coefficients). Problems at page 161
Problem number : 39
Date solved : Monday, January 27, 2025 at 02:46:48 AM
CAS classification : [[_3rd_order, _missing_y]]

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

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 27

dsolve([diff(y(x),x$3)+diff(y(x),x$2)=x+exp(-x),y(0) = 1, D(y)(0) = 0, (D@@2)(y)(0) = 1],y(x), singsol=all)
 

\[ y = -3+\left (x +4\right ) {\mathrm e}^{-x}+\frac {x^{3}}{6}-\frac {x^{2}}{2}+3 x \]

✓ Solution by Mathematica

Time used: 0.223 (sec). Leaf size: 32

DSolve[{D[y[x],{x,3}]+D[y[x],{x,2}]==x+Exp[-x],{y[0]==1,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{6} \left (x^3-3 x^2+12 x-12\right )+e^{-x} (x+3) \]