A lightly stretched homogeneous string of length \( L \), with its fixed ends at \( x=0 \) and \( x=L \) executes transverse vibration. Motion is started with zero initial velocity by displacing the string into the form \( f(x)=k\left(x^{2}-x^{3}\right) \). Find the deflection \( u(x, t) \)

The motion of a particle performing damped vibrations is given by \( \gamma=e^{-t} \sin 2 t, \gamma \) being the displacement trom its mean position at time \( t \). Show that \( \gamma \) is a maximum when \( t=\frac{1}{2 \tan ^{-1}(2)} \) and determine this maximum displacement to three significant figures.

The motion of a particle performing damped vibrations is given by \( \gamma=e^{-t} \operatorname{Sin} 2 t, \gamma \) being the displacement trom its mean position at time \( t \). Show that \( \gamma \) is a maximum when \( t=1 / 2 \tan ^{-1}(2) \) and determine this maximum displacement to three significant figures.

50. Determinaţi valorile parametrului real \( m \) pentru care graficul funcțici \( f: R \rightarrow R \), \( f(x)=\sqrt[3]{x^{2}+m x+1} \) are tangentă in fiecare punct \( x \in R \).

\( \int ( 2 x + 3 ) ^ { 2 } \sqrt[ 3 ] { x - 2 } \)

The motion of a particle pertorming damped vibrations s given by \( \gamma=e^{-t} \operatorname{Sin} 2 t, \gamma \) being the displacement rom its mean position at time \( t \). Show that \( \gamma \) is a maximum when \( t=\frac{1}{2} \tan ^{-1}(2) \) and determine this maximum displacement to hree significant tigures.