13.8.9 problem 9

Internal problem ID [2384]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2.1, Complex roots. Page 141
Problem number : 9
Date solved : Monday, January 27, 2025 at 05:51:06 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 3 y^{\prime \prime }-2 y^{\prime }+4 y&=0 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.216 (sec). Leaf size: 70

dsolve([3*diff(y(t),t$2)-2*diff(y(t),t)+4*y(t)=0,y(2) = 1, D(y)(2) = -1],y(t), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-\frac {2}{3}+\frac {t}{3}} \left (\left (\sin \left (\frac {2 \sqrt {11}}{3}\right ) \sqrt {11}-4 \cos \left (\frac {2 \sqrt {11}}{3}\right )\right ) \sin \left (\frac {\sqrt {11}\, t}{3}\right )+\left (\cos \left (\frac {2 \sqrt {11}}{3}\right ) \sqrt {11}+4 \sin \left (\frac {2 \sqrt {11}}{3}\right )\right ) \cos \left (\frac {\sqrt {11}\, t}{3}\right )\right ) \sqrt {11}}{11} \]

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 54

DSolve[{3*D[y[t],{t,2}]-2*D[y[t],t]+4*y[t]==0,{y[2]==1,Derivative[1][y][2]==-1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{11} e^{\frac {t-2}{3}} \left (11 \cos \left (\frac {1}{3} \sqrt {11} (t-2)\right )-4 \sqrt {11} \sin \left (\frac {1}{3} \sqrt {11} (t-2)\right )\right ) \]