15.11.16 problem 16

Internal problem ID [3126]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 19, page 86
Problem number : 16
Date solved : Sunday, March 30, 2025 at 01:19:03 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }-3 y^{\prime \prime }+4 y^{\prime }-12 y&=x +{\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 33

ode:=diff(diff(diff(y(x),x),x),x)-3*diff(diff(y(x),x),x)+4*diff(y(x),x)-12*y(x) = x+exp(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{2 x}}{8}-\frac {x}{12}-\frac {1}{36}+c_1 \cos \left (2 x \right )+c_2 \,{\mathrm e}^{3 x}+c_3 \sin \left (2 x \right ) \]

✓ Mathematica. Time used: 0.426 (sec). Leaf size: 45

ode=D[y[x],{x,3}]-3*D[y[x],{x,2}]+4*D[y[x],x]-12*y[x]==x+Exp[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{72} \left (-6 x-9 e^{2 x}+72 c_3 e^{3 x}-2\right )+c_1 \cos (2 x)+c_2 \sin (2 x) \]

✓ Sympy. Time used: 0.243 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x - 12*y(x) - exp(2*x) + 4*Derivative(y(x), x) - 3*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{3 x} + C_{2} \sin {\left (2 x \right )} + C_{3} \cos {\left (2 x \right )} - \frac {x}{12} - \frac {e^{2 x}}{8} - \frac {1}{36} \]