31.1.10 problem 10.2.11 (iv)

Internal problem ID [7685]
Book : Basic Training in Mathematics. By R. Shankar. Plenum Press. NY. 1995
Section : Chapter 10, Differential equations. Section 10.2, ODEs with constant Coefficients. page 307
Problem number : 10.2.11 (iv)
Date solved : Tuesday, September 30, 2025 at 04:55:55 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 12

ode:=diff(diff(y(x),x),x)-y(x) = cosh(x); 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \cosh \left (x \right )+\frac {x \sinh \left (x \right )}{2} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 28

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

\begin{align*} y(x)&\to \frac {1}{4} e^{-x} \left (-x+e^{2 x} (x+2)+2\right ) \end{align*}

✓ Sympy. Time used: 0.060 (sec). Leaf size: 20

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

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