75.18.16 problem 605

Internal problem ID [17104]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.3 Nonhomogeneous linear equations with constant coefficients. Initial value problem. Exercises page 140
Problem number : 605
Date solved : Tuesday, January 28, 2025 at 09:52:49 AM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime \prime }-y&=8 \,{\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\\ y^{\prime \prime \prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.041 (sec). Leaf size: 23

dsolve([diff(y(x),x$4)-y(x)=8*exp(x),y(0) = -1, D(y)(0) = 0, (D@@2)(y)(0) = 1, (D@@3)(y)(0) = 0],y(x), singsol=all)
 

\[ y = {\mathrm e}^{-x}+\left (2 x -3\right ) {\mathrm e}^{x}+\cos \left (x \right )+2 \sin \left (x \right ) \]

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 104

DSolve[{D[y[x],{x,4}]-y[x]==8*Exp[x],{y[0]==-1,Derivative[1][y][0] ==0,Derivative[2][y][0] ==1,Derivative[3][y][0]==0}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\sin (x) \int _1^0-4 e^{K[2]} \cos (K[2])dK[2]+\sin (x) \int _1^x-4 e^{K[2]} \cos (K[2])dK[2]-\cos (x) \int _1^04 e^{K[1]} \sin (K[1])dK[1]+\cos (x) \int _1^x4 e^{K[1]} \sin (K[1])dK[1]+2 e^x x+e^{-x}-e^x-\cos (x) \]