69.17.4 problem 554

Internal problem ID [18340]
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 : 554
Date solved : Thursday, October 02, 2025 at 03:10:37 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+2 y&=\left (1+\sin \left (x \right )\right ) {\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 26

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)+2*y(x) = (1+sin(x))*exp(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{x} \left (\left (x -2 c_1 \right ) \cos \left (x \right )-2+\left (-2 c_2 -1\right ) \sin \left (x \right )\right )}{2} \]

✓ Mathematica. Time used: 0.124 (sec). Leaf size: 71

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

\begin{align*} y(x)&\to \frac {1}{4} e^x \left (4 \cos (x) \int _1^x-\left (\cos \left (\frac {K[1]}{2}\right )+\sin \left (\frac {K[1]}{2}\right )\right )^2 \sin (K[1])dK[1]+4 c_2 \cos (x)+\sin (x) (4 \sin (x)-\cos (2 x)+3+4 c_1)\right ) \end{align*}

✓ Sympy. Time used: 0.186 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-sin(x) - 1)*exp(x) + 2*y(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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