58.2.42 problem 42

Internal problem ID [9165]
Book : Second order enumerated odes
Section : section 2
Problem number : 42
Date solved : Sunday, March 30, 2025 at 02:24:35 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-3 y^{\prime \prime }+5 y^{\prime }-2 y&=x \,{\mathrm e}^{x}+3 \,{\mathrm e}^{-2 x} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 52

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

\[ y = \frac {\left (-216 x +1944 c_2 -216\right ) {\mathrm e}^{-2 x}}{1944}+\frac {{\mathrm e}^{x} \left (x^{4}-\frac {4 x^{3}}{3}+\left (72 c_4 +\frac {4}{3}\right ) x^{2}+\left (72 c_3 -\frac {8}{9}\right ) x +72 c_1 +\frac {8}{27}\right )}{72} \]

✓ Mathematica. Time used: 0.279 (sec). Leaf size: 170

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

\[ y(x)\to e^x x \int _1^x-\frac {1}{9} e^{-3 K[3]} (3 K[3]+1) \left (e^{3 K[3]} K[3]+3\right )dK[3]+e^{-2 x} \int _1^x\left (-\frac {1}{27} e^{3 K[1]} K[1]-\frac {1}{9}\right )dK[1]+e^x \int _1^x\frac {1}{54} e^{-3 K[2]} \left (e^{3 K[2]} K[2]+3\right ) \left (9 K[2]^2+6 K[2]+2\right )dK[2]+\frac {e^x x^4}{12}-\frac {1}{6} e^{-2 x} x^2+c_4 e^x x^2+c_3 e^x x+c_1 e^{-2 x}+c_2 e^x \]

✓ Sympy. Time used: 0.475 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = \left (C_{1} - \frac {x}{9}\right ) e^{- 2 x} + \left (C_{2} + x \left (C_{3} + x \left (C_{4} + \frac {x^{2}}{72} - \frac {x}{54}\right )\right )\right ) e^{x} \]