8.11.20 problem 35

Internal problem ID [888]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.5, Nonhomogeneous equations and undetermined coefficients Page 351
Problem number : 35
Date solved : Tuesday, March 04, 2025 at 11:58:13 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+2 y&=x +1 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 22

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)+2*y(x) = 1+x; 
ic:=y(0) = 3, D(y)(0) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {\left (4 \cos \left (x \right )-5 \sin \left (x \right )\right ) {\mathrm e}^{x}}{2}+\frac {x}{2}+1 \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 26

ode=D[y[x],{x,2}]-2*D[y[x],x]+2*y[x]==x+1; 
ic={y[0]==3,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{2} \left (x-5 e^x \sin (x)+4 e^x \cos (x)+2\right ) \]

✓ Sympy. Time used: 0.184 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x + 2*y(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 1,0) 
ics = {y(0): 3, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x}{2} + \left (- \frac {5 \sin {\left (x \right )}}{2} + 2 \cos {\left (x \right )}\right ) e^{x} + 1 \]