15.11.27 problem 27

Internal problem ID [3137]
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 : 27
Date solved : Sunday, March 30, 2025 at 01:19:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-5 y^{\prime }-6 y&={\mathrm e}^{3 x} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 23

ode:=diff(diff(y(x),x),x)-5*diff(y(x),x)-6*y(x) = exp(3*x); 
ic:=y(0) = 2, D(y)(0) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {45 \,{\mathrm e}^{-x}}{28}+\frac {10 \,{\mathrm e}^{6 x}}{21}-\frac {{\mathrm e}^{3 x}}{12} \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 30

ode=D[y[x],{x,2}]-5*D[y[x],x]-6*y[x]==Exp[3*x]; 
ic={y[0]==2,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{84} e^{-x} \left (-7 e^{4 x}+40 e^{7 x}+135\right ) \]

✓ Sympy. Time used: 0.204 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-6*y(x) - exp(3*x) - 5*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {10 e^{6 x}}{21} - \frac {e^{3 x}}{12} + \frac {45 e^{- x}}{28} \]