71.13.13 problem 13

Internal problem ID [14454]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 5. The Laplace Transform Method. Exercises 5.2, page 248
Problem number : 13
Date solved : Thursday, March 13, 2025 at 03:30:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 8.183 (sec). Leaf size: 18

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)+2*y(x) = -x^2+1; 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(x),method='laplace');
 

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

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 20

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

\[ y(x)\to e^x \cos (x)-\frac {1}{2} x (x+2) \]

✓ Sympy. Time used: 0.206 (sec). Leaf size: 15

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

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