75.20.10 problem 645

Internal problem ID [17144]
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 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 645
Date solved : Tuesday, January 28, 2025 at 09:54:10 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+{\mathrm e}^{-2 x} y&={\mathrm e}^{-3 x} \end{align*}

Using reduction of order method given that one solution is

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 28

dsolve([diff(y(x),x$2)+diff(y(x),x)+exp(-2*x)*y(x)=exp(-3*x),cos(exp(-x))],singsol=all)
 

\[ y = \sin \left ({\mathrm e}^{-x}\right ) c_{2} +\cos \left ({\mathrm e}^{-x}\right ) c_{1} +\sin \left ({\mathrm e}^{-x}\right )+{\mathrm e}^{-x} \]

✓ Solution by Mathematica

Time used: 0.066 (sec). Leaf size: 82

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

\[ y(x)\to \cos \left (e^{-x}\right ) \int _1^xe^{-2 K[1]} \sin \left (e^{-K[1]}\right )dK[1]-\sin \left (e^{-x}\right ) \int _1^xe^{-2 K[2]} \cos \left (e^{-K[2]}\right )dK[2]+c_1 \cos \left (e^{-x}\right )-c_2 \sin \left (e^{-x}\right ) \]