64.11.50 problem 50

Internal problem ID [13429]
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 : 50
Date solved : Monday, March 31, 2025 at 07:56:59 AM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\left (6\right )}+2 y^{\left (5\right )}+5 y^{\prime \prime \prime \prime }&=x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right ) \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 91

ode:=diff(diff(diff(diff(diff(diff(y(x),x),x),x),x),x),x)+2*diff(diff(diff(diff(diff(y(x),x),x),x),x),x)+5*diff(diff(diff(diff(y(x),x),x),x),x) = x^3+x^2*exp(-x)+exp(-x)*sin(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-483-219 i+\left (70-240 i\right ) x +\left (-280+960 i\right ) c_1 +\left (-960-280 i\right ) c_2 \right ) {\mathrm e}^{\left (-1-2 i\right ) x}}{50000}+\frac {\left (-483+219 i+\left (70+240 i\right ) x +\left (-280-960 i\right ) c_1 +\left (-960+280 i\right ) c_2 \right ) {\mathrm e}^{\left (-1+2 i\right ) x}}{50000}+\frac {\left (2 x^{2}+16 x +39\right ) {\mathrm e}^{-x}}{8}+\frac {x^{7}}{4200}-\frac {x^{6}}{1500}-\frac {x^{5}}{2500}+\frac {3 x^{4}}{625}+\frac {c_3 \,x^{3}}{6}+\frac {c_4 \,x^{2}}{2}+c_5 x +c_6 \]

✓ Mathematica. Time used: 60.208 (sec). Leaf size: 161

ode=D[y[x],{x,6}]+2*D[y[x],{x,5}]+5*D[y[x],{x,4}]==x^3+x^2*Exp[-x]+Exp[-x]*Sin[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \int _1^x\int _1^{K[6]}\int _1^{K[5]}\int _1^{K[4]}e^{-K[3]} \left (c_2 \cos (2 K[3])+\int _1^{K[3]}-\frac {1}{2} \sin (2 K[2]) \left (e^{K[2]} K[2]^3+K[2]^2+\sin (2 K[2])\right )dK[2] \cos (2 K[3])+c_1 \sin (2 K[3])+\sin (2 K[3]) \int _1^{K[3]}\frac {1}{2} \cos (2 K[1]) \left (e^{K[1]} K[1]^3+K[1]^2+\sin (2 K[1])\right )dK[1]\right )dK[3]dK[4]dK[5]dK[6]+x (x (c_6 x+c_5)+c_4)+c_3 \]

✓ Sympy. Time used: 0.489 (sec). Leaf size: 88

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3 - x**2*exp(-x) + 5*Derivative(y(x), (x, 4)) + 2*Derivative(y(x), (x, 5)) + Derivative(y(x), (x, 6)) - exp(-x)*sin(2*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + C_{2} x^{3} + \frac {x^{7}}{4200} - \frac {x^{6}}{1500} - \frac {x^{5}}{2500} + \frac {3 x^{4}}{625} + x^{2} \left (C_{4} + \frac {e^{- x}}{4}\right ) + x \left (C_{3} + \left (- \frac {6 \sin {\left (2 x \right )}}{625} + \frac {7 \cos {\left (2 x \right )}}{2500} + 2\right ) e^{- x}\right ) + \left (C_{5} \sin {\left (2 x \right )} + C_{6} \cos {\left (2 x \right )} + \frac {39}{8}\right ) e^{- x} \]