69.17.28 problem 578

Internal problem ID [18364]
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 : 578
Date solved : Thursday, October 02, 2025 at 03:10:56 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+5 y&={\mathrm e}^{x} \left (1-2 \sin \left (x \right )^{2}\right )+10 x +1 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 30

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)+5*y(x) = exp(x)*(1-2*sin(x)^2)+10*x+1; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{x} \left (x +4 c_2 \right ) \sin \left (2 x \right )}{4}+{\mathrm e}^{x} \cos \left (2 x \right ) c_1 +2 x +1 \]

✓ Mathematica. Time used: 0.434 (sec). Leaf size: 110

ode=D[y[x],{x,2}]-2*D[y[x],x]+5*y[x]==Exp[x]*(1-2*Sin[x]^2)+10*x+1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^x \left (\cos (2 x) \int _1^x-\frac {1}{2} e^{-K[2]} \left (e^{K[2]} \cos (2 K[2])+10 K[2]+1\right ) \sin (2 K[2])dK[2]+\sin (2 x) \int _1^x\frac {1}{2} e^{-K[1]} \cos (2 K[1]) \left (e^{K[1]} \cos (2 K[1])+10 K[1]+1\right )dK[1]+c_2 \cos (2 x)+c_1 \sin (2 x)\right ) \end{align*}

✓ Sympy. Time used: 20.521 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-10*x - (1 - 2*sin(x)**2)*exp(x) + 5*y(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = 2 x + \left (C_{2} \cos {\left (2 x \right )} + \left (C_{1} + \frac {x}{4}\right ) \sin {\left (2 x \right )}\right ) e^{x} + 1 \]