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 : Monday, January 27, 2025 at 04:50:08 AM
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*}

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

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

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