75.18.13 problem 602

Internal problem ID [17101]
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 : 602
Date solved : Tuesday, January 28, 2025 at 09:52:38 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }-y^{\prime }&=-5 \,{\mathrm e}^{-x} \left (\sin \left (x \right )+\cos \left (x \right )\right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-4\\ y^{\prime }\left (0\right )&=5 \end{align*}

✓ Solution by Maple

Time used: 0.035 (sec). Leaf size: 22

dsolve([diff(y(x),x$2)-diff(y(x),x)=-5*exp(-x)*(sin(x)+cos(x)),y(0) = -4, D(y)(0) = 5],y(x), singsol=all)
 

\[ y = -4+2 \,{\mathrm e}^{x}+{\mathrm e}^{-x} \left (\sin \left (x \right )-2 \cos \left (x \right )\right ) \]

✓ Solution by Mathematica

Time used: 7.972 (sec). Leaf size: 131

DSolve[{D[y[x],{x,2}]-D[y[x],x]==-5*Exp[-x]*(Sin[x]+Cos[x]),{y[0]==-4,Derivative[1][y][0] ==5}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \int _1^xe^{K[2]} \left (-\int _1^0-5 e^{-2 K[1]} (\cos (K[1])+\sin (K[1]))dK[1]+\int _1^{K[2]}-5 e^{-2 K[1]} (\cos (K[1])+\sin (K[1]))dK[1]+5\right )dK[2]-\int _1^0e^{K[2]} \left (-\int _1^0-5 e^{-2 K[1]} (\cos (K[1])+\sin (K[1]))dK[1]+\int _1^{K[2]}-5 e^{-2 K[1]} (\cos (K[1])+\sin (K[1]))dK[1]+5\right )dK[2]-4 \]