15.14.20 problem 20

Internal problem ID [3192]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 23, page 106
Problem number : 20
Date solved : Tuesday, March 04, 2025 at 04:05:42 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }+4 y^{\prime \prime }-5 y^{\prime }&=x^{2} {\mathrm e}^{-x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 41

ode:=diff(diff(diff(y(x),x),x),x)+4*diff(diff(y(x),x),x)-5*diff(y(x),x) = x^2*exp(-x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-5 x} \left (\left (x^{2}+\frac {5}{2} x +\frac {23}{8}\right ) {\mathrm e}^{4 x}+8 c_{1} {\mathrm e}^{6 x}+8 c_3 \,{\mathrm e}^{5 x}-\frac {8 c_2}{5}\right )}{8} \]

✓ Mathematica. Time used: 0.289 (sec). Leaf size: 44

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

\[ y(x)\to \frac {1}{64} e^{-x} \left (8 x^2+20 x+23\right )-\frac {1}{5} c_1 e^{-5 x}+c_2 e^x+c_3 \]

✓ Sympy. Time used: 0.310 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{1} + C_{2} e^{- 5 x} + C_{3} e^{x} + \frac {\left (8 x^{2} + 20 x + 23\right ) e^{- x}}{64} \]