64.11.53 problem 53

Internal problem ID [13424]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.3. The method of undetermined coefficients. Exercises page 151
Problem number : 53
Date solved : Wednesday, March 05, 2025 at 09:57:58 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+3 y^{\prime \prime }-4 y&=\cos \left (x \right )^{2}-\cosh \left (x \right ) \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 56

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+3*diff(diff(y(x),x),x)-4*y(x) = cos(x)^2-cosh(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {1}{8}+\frac {\left (10 x +200 c_{3} +9\right ) {\mathrm e}^{-x}}{200}+\frac {\left (200 c_{2} -9\right ) \cos \left (2 x \right )}{200}+\frac {\left (-x +40 c_4 \right ) \sin \left (2 x \right )}{40}+\frac {\left (-10 x +200 c_{1} +9\right ) {\mathrm e}^{x}}{200} \]

✓ Mathematica. Time used: 1.252 (sec). Leaf size: 286

ode=D[y[x],{x,4}]+3*D[y[x],{x,2}]-4*y[x]==Cos[x]^2-Cosh[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-x} \int _1^x\frac {1}{20} \left (-e^{K[3]} \cos (2 K[3])-e^{K[3]}+e^{2 K[3]}+1\right )dK[3]+\sin (2 x) \int _1^x-\frac {1}{20} \cos (2 K[2]) (\cos (2 K[2])-2 \cosh (K[2])+1)dK[2]+\cos (2 x) \int _1^x\frac {1}{20} (\cos (2 K[1])-2 \cosh (K[1])+1) \sin (2 K[1])dK[1]+c_3 e^{-x}+c_4 e^x+c_1 \cos (2 x)+c_2 \sin (2 x)-\frac {(\cos (2 x)-2 \cosh (x)+1) \left (10 x (1-i \tan (x))^i-5 (1+i \tan (x))^i+10 \sec ^2(x)^{1 i/2}+2 \cos (2 x) \sec ^2(x)^{1 i/2}-4 \sin (2 x) \sec ^2(x)^{1 i/2}\right )}{200 \left (-(1-i \tan (x))^i-(1+i \tan (x))^i+\sec ^2(x)^{1 i/2}+\cos (2 x) \sec ^2(x)^{1 i/2}\right )} \]

✓ Sympy. Time used: 2.679 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*y(x) - cos(x)**2 + cosh(x) + 3*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{2} e^{- x} + C_{3} e^{x} + C_{4} \cos {\left (2 x \right )} - \frac {x \sinh {\left (x \right )}}{10} + \left (C_{1} - \frac {x}{40}\right ) \sin {\left (2 x \right )} + \frac {7 \cosh {\left (x \right )}}{50} - \frac {1}{8} \]