66.2.26 problem Problem 35

Internal problem ID [13854]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER. Problems page 172
Problem number : Problem 35
Date solved : Monday, March 31, 2025 at 08:15:17 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 28

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

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

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 37

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

\[ y(x)\to \frac {1}{8} e^x \left (\left (2 x^2-1+8 c_1\right ) \sin (x)+2 (x+4 c_2) \cos (x)\right ) \]

✓ Sympy. Time used: 0.309 (sec). Leaf size: 24

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

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