44.14.14 problem 2(f)

Internal problem ID [9338]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Problems for Review and Discovery. Drill excercises. Page 105
Problem number : 2(f)
Date solved : Tuesday, September 30, 2025 at 06:16:35 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=\cos \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=3 \\ y^{\prime }\left (2\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.243 (sec). Leaf size: 27

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

\[ y = -\frac {\cos \left (x \right )}{2}-\frac {\left (\sin \left (2\right )-4\right ) \sinh \left (x \right ) \operatorname {sech}\left (2\right )}{2}+\frac {7 \cosh \left (x -2\right ) \operatorname {sech}\left (2\right )}{2} \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 294

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

\begin{align*} y(x)&\to \frac {e^{-x} \left (-\left (\left (e^{2 x}+e^4\right ) \int _1^0\frac {1}{2} e^{-K[1]} \cos (K[1])dK[1]\right )-e^4 \left (e^{2 x}-1\right ) \int _1^2\frac {1}{2} e^{-K[1]} \cos (K[1])dK[1]+e^{2 x} \int _1^x\frac {1}{2} e^{-K[1]} \cos (K[1])dK[1]+e^{2 x+4} \int _1^x\frac {1}{2} e^{-K[1]} \cos (K[1])dK[1]-e^{2 x} \int _1^0-\frac {1}{2} e^{K[2]} \cos (K[2])dK[2]+e^{2 x} \int _1^2-\frac {1}{2} e^{K[2]} \cos (K[2])dK[2]+e^4 \int _1^x-\frac {1}{2} e^{K[2]} \cos (K[2])dK[2]+\int _1^x-\frac {1}{2} e^{K[2]} \cos (K[2])dK[2]-e^4 \int _1^0-\frac {1}{2} e^{K[2]} \cos (K[2])dK[2]-\int _1^2-\frac {1}{2} e^{K[2]} \cos (K[2])dK[2]+3 e^{2 x}+2 e^{2 x+2}+3 e^4-2 e^2\right )}{1+e^4} \end{align*}

✓ Sympy. Time used: 0.064 (sec). Leaf size: 58

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

\[ y{\left (x \right )} = \frac {\left (- e^{2} \sin {\left (2 \right )} + 7 + 4 e^{2}\right ) e^{x}}{2 + 2 e^{4}} - \frac {\cos {\left (x \right )}}{2} + \frac {\left (- 4 e^{2} + e^{2} \sin {\left (2 \right )} + 7 e^{4}\right ) e^{- x}}{2 + 2 e^{4}} \]