Internal problem ID [15414]
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 ODE’s). Section 15.5 Linear equations with variable coefficients.
The Lagrange method. Exercises page 148
Problem number: 645.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+y^{\prime }+{\mathrm e}^{-2 x} y={\mathrm e}^{-3 x}} \] Given that one solution of the ode is \begin {align*} y_1 &= \cos \left ({\mathrm e}^{-x}\right ) \end {align*}
✓ Solution by Maple
Time used: 0.016 (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)
\[ y \left (x \right ) = \sin \left ({\mathrm e}^{-x}\right ) c_{2} +\cos \left ({\mathrm e}^{-x}\right ) c_{1} +\sin \left ({\mathrm e}^{-x}\right )+{\mathrm e}^{-x} \]
✓ Solution by Mathematica
Time used: 0.131 (sec). Leaf size: 30
DSolve[y''[x]+y'[x]+Exp[-2*x]*y[x]==Exp[-3*x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-x}+c_1 \cos \left (e^{-x}\right )-c_2 \sin \left (e^{-x}\right ) \]