69.18.13 problem 602

Internal problem ID [18388]
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 : Thursday, October 02, 2025 at 03:11:15 PM
CAS classification : [[_2nd_order, _missing_y]]

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

With initial conditions

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 22

ode:=diff(diff(y(x),x),x)-diff(y(x),x) = -5*exp(-x)*(cos(x)+sin(x)); 
ic:=[y(0) = -4, D(y)(0) = 5]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 5.123 (sec). Leaf size: 131

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

\begin{align*} 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 \end{align*}

✓ Sympy. Time used: 0.217 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((5*sin(x) + 5*cos(x))*exp(-x) - Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): -4, Subs(Derivative(y(x), x), x, 0): 5} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\sqrt {2} \left (- \sin {\left (x + \frac {\pi }{4} \right )} - 3 \cos {\left (x + \frac {\pi }{4} \right )}\right ) e^{- x}}{2} + 2 e^{x} - 4 \]