13.8.8 problem 8

Internal problem ID [2383]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2.1, Complex roots. Page 141
Problem number : 8
Date solved : Monday, January 27, 2025 at 05:51:01 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 y^{\prime \prime }-y^{\prime }+3 y&=0 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.243 (sec). Leaf size: 70

dsolve([2*diff(y(t),t$2)-diff(y(t),t)+3*y(t)=0,y(1) = 1, D(y)(1) = 1],y(t), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-\frac {1}{4}+\frac {t}{4}} \left (\left (\sin \left (\frac {\sqrt {23}}{4}\right ) \sqrt {23}+3 \cos \left (\frac {\sqrt {23}}{4}\right )\right ) \sin \left (\frac {\sqrt {23}\, t}{4}\right )+\left (\cos \left (\frac {\sqrt {23}}{4}\right ) \sqrt {23}-3 \sin \left (\frac {\sqrt {23}}{4}\right )\right ) \cos \left (\frac {\sqrt {23}\, t}{4}\right )\right ) \sqrt {23}}{23} \]

✓ Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 54

DSolve[{2*D[y[t],{t,2}]-D[y[t],t]+3*y[t]==0,{y[1]==1,Derivative[1][y][1]==1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{23} e^{\frac {t-1}{4}} \left (3 \sqrt {23} \sin \left (\frac {1}{4} \sqrt {23} (t-1)\right )+23 \cos \left (\frac {1}{4} \sqrt {23} (t-1)\right )\right ) \]