76.13.39 problem 39

Internal problem ID [17550]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.3 (Linear homogeneous equations with constant coefficients). Problems at page 239
Problem number : 39
Date solved : Monday, March 31, 2025 at 04:17:10 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+y^{\prime }+\frac {5 y}{4}&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.066 (sec). Leaf size: 18

ode:=diff(diff(y(x),x),x)+diff(y(x),x)+5/4*y(x) = 0; 
ic:=y(0) = 3, D(y)(0) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {x}{2}} \left (\frac {5 \sin \left (x \right )}{2}+3 \cos \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 25

ode=D[y[x],{x,2}]+D[y[x],x]+125/100*y[x]==0; 
ic={y[0]==3,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{2} e^{-x/2} (5 \sin (x)+6 \cos (x)) \]

✓ Sympy. Time used: 0.144 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(5*y(x)/4 + Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {5 \sin {\left (x \right )}}{2} + 3 \cos {\left (x \right )}\right ) e^{- \frac {x}{2}} \]