70.13.42 problem 42

Internal problem ID [18911]
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 : 42
Date solved : Thursday, October 02, 2025 at 03:32:45 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+2 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (\frac {\pi }{4}\right )&=2 \\ y^{\prime }\left (\frac {\pi }{4}\right )&=-2 \\ \end{align*}

✓ Maple. Time used: 0.107 (sec). Leaf size: 21

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)+2*y(x) = 0; 
ic:=[y(1/4*Pi) = 2, D(y)(1/4*Pi) = -2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \sqrt {2}\, {\mathrm e}^{-x +\frac {\pi }{4}} \left (\sin \left (x \right )+\cos \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 27

ode=D[y[x],{x,2}]+2*D[y[x],x]+2*y[x]==0; 
ic={y[Pi/4]==2,Derivative[1][y][Pi/4] ==-2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \sqrt {2} e^{\frac {\pi }{4}-x} (\sin (x)+\cos (x)) \end{align*}

✓ Sympy. Time used: 0.105 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = \left (\sqrt {2} e^{\frac {\pi }{4}} \sin {\left (x \right )} + \sqrt {2} e^{\frac {\pi }{4}} \cos {\left (x \right )}\right ) e^{- x} \]