76.13.42 problem 42

Internal problem ID [17624]
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 : Tuesday, January 28, 2025 at 10:45:52 AM
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*}

✓ Solution by Maple

Time used: 0.093 (sec). Leaf size: 21

dsolve([diff(y(x),x$2)+2*diff(y(x),x)+2*y(x)=0,y(1/4*Pi) = 2, D(y)(1/4*Pi) = -2],y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.014 (sec). Leaf size: 27

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

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