8.11.7 problem 7

Internal problem ID [875]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.5, Nonhomogeneous equations and undetermined coefficients Page 351
Problem number : 7
Date solved : Tuesday, March 04, 2025 at 11:56:58 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y&=\sinh \left (x \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 56

ode:=diff(diff(y(x),x),x)-4*y(x) = sinh(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-2 \sinh \left (x \right )^{2} \cosh \left (x \right )-2 \sinh \left (x \right )^{3}+12 c_1 +\cosh \left (x \right )\right ) {\mathrm e}^{-2 x}}{12}+\left (\frac {\sinh \left (x \right )^{2} \cosh \left (x \right )}{6}-\frac {\sinh \left (x \right )^{3}}{6}+c_2 -\frac {\cosh \left (x \right )}{12}\right ) {\mathrm e}^{2 x} \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 38

ode=D[y[x],{x,2}]-4*y[x]==Sinh[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{6} e^{-2 x} \left (e^x-e^{3 x}+6 c_1 e^{4 x}+6 c_2\right ) \]

✓ Sympy. Time used: 0.084 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*y(x) - sinh(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{- 2 x} + C_{2} e^{2 x} - \frac {\sinh {\left (x \right )}}{3} \]