10.8.20 problem 26

Internal problem ID [1292]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 3, Second order linear equations, 3.3 Complex Roots of the Characteristic Equation , page 164
Problem number : 26
Date solved : Tuesday, March 04, 2025 at 12:28:11 PM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 17

ode:=diff(diff(y(x),x),x)+2*a*diff(y(x),x)+(a^2+1)*y(x) = 0; 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = {\mathrm e}^{-a x} \left (a \sin \left (x \right )+\cos \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 94

ode=D[y[x],{x,2}]+2*a*D[y[x],x]+(a^1+1)*y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {e^{-\left (\left (\sqrt {a^2-a-1}+a\right ) x\right )} \left (a \left (e^{2 \sqrt {a^2-a-1} x}-1\right )+\sqrt {a^2-a-1} \left (e^{2 \sqrt {a^2-a-1} x}+1\right )\right )}{2 \sqrt {a^2-a-1}} \]

✓ Sympy. Time used: 0.231 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = \left (- \frac {i a}{2} + \frac {1}{2}\right ) e^{x \left (- a + i\right )} + \left (\frac {i a}{2} + \frac {1}{2}\right ) e^{- x \left (a + i\right )} \]