75.17.11 problem 561

Internal problem ID [16989]
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. Superposition principle. Exercises page 137
Problem number : 561
Date solved : Monday, March 31, 2025 at 03:37:19 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+y&=2+{\mathrm e}^{x} \sin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 18

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)+y(x) = 2+exp(x)*sin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = 2+\left (c_1 x +c_2 -\sin \left (x \right )\right ) {\mathrm e}^{x} \]

✓ Mathematica. Time used: 0.216 (sec). Leaf size: 61

ode=D[y[x],{x,2}]-2*D[y[x],x]+y[x]==2+Exp[x]*Sin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.216 (sec). Leaf size: 15

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

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