75.18.12 problem 601

Internal problem ID [17100]
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 : 601
Date solved : Tuesday, January 28, 2025 at 09:52:36 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-6 y^{\prime }+9 y&=16 \,{\mathrm e}^{-x}+9 x -6 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.073 (sec). Leaf size: 16

dsolve([diff(y(x),x$2)-6*diff(y(x),x)+9*y(x)=16*exp(-x)+9*x-6,y(0) = 1, D(y)(0) = 1],y(x), singsol=all)
 

\[ y = {\mathrm e}^{3 x} x +x +{\mathrm e}^{-x} \]

✓ Solution by Mathematica

Time used: 0.393 (sec). Leaf size: 137

DSolve[{D[y[x],{x,2}]-6*D[y[x],x]+9*y[x]==16*Exp[-x]+9*x-6,{y[0]==1,Derivative[1][y][0] ==1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{3 x} \left (x \left (-\int _1^0e^{-4 K[2]} \left (e^{K[2]} (9 K[2]-6)+16\right )dK[2]\right )+x \int _1^xe^{-4 K[2]} \left (e^{K[2]} (9 K[2]-6)+16\right )dK[2]+\int _1^x-e^{-4 K[1]} K[1] \left (e^{K[1]} (9 K[1]-6)+16\right )dK[1]-\int _1^0-e^{-4 K[1]} K[1] \left (e^{K[1]} (9 K[1]-6)+16\right )dK[1]-2 x+1\right ) \]