Solve the problem of un-damped forced vibrations of a spring in the case where the forcing function is f(t)=A sin t. D.E associated with the problem is (m frac{d^2 y}{dt^2} + ky = f(t) ), with initial conditions as y(0)=y0 and y (0)=y1 and assume 2 = k/m, =A/m.

y = y0 cos u2061 t + y1 sin u2061 t + ( frac{ cos t}{- ^2+ ^2} )

y = y0 cos u2061 t + (y1/ )sin u2061 t + ( frac{cos t}{ ^2+ ^2} )

y = y0 cos u2061 t + (y1 )sin u2061 t + ( frac{sin t}{ ^2+ ^2} )

y = y0 cos u2061 t + (y1/ )sin u2061 t + ( frac{ sin t}{- ^2+ ^2} )

Solve the problem of resonance damped vibration of a spring .If the governing D.E is given by (m frac{d^2 y}{dt^2} + c frac{dy}{dt} + ky=0; ) c>0 with initial conditions as y(0)=y0 and y (0)=y1 and assume c/m=2 , k/m= 2 and (v = sqrt{ ^2- ^2} ).

y = e^(- t) ( ( left( frac{y_1+ y_0}{v} right) ) cos u2061vt + y0 sin u2061vt)

y = e- t (y0 cos u2061vt + y1 sin u2061vt)

y = e^(- t) (y0 cos u2061vt+ ( left( frac{y_1+ y_0}{v} right) ) sin u2061vt)

y = e^(- t) ( ( left( frac{y_1+ y_0}{v} right) ) cos u2061vt + y0 sin u2061vt)

y = e- t (y0 cos u2061vt+ y1 sin u2061vt)

A particle moves along the x-axis according to the law ( frac{d^2 x}{dt^2} + 6 frac{dx}{dt} + 25x = 0. ) If the particle is started at x=0 with an initial velocity of 12ft/sec to the left, determine x in terms of t.

x=-3e-3t (sin u20614t+cos u20614t)

Explore topic-wise MCQs in Ordinary Differential Equations.

This section includes 4 Mcqs, each offering curated multiple-choice questions to sharpen your Ordinary Differential Equations knowledge and support exam preparation. Choose a topic below to get started.

1.	A particle undergoes forced vibrations according to the law x (t) + 25x(t) = 21 sin t. If the particle starts from rest at t=0, find the displacement at any time t>0.
A.	( frac{21 cos t}{25} )
B.	( frac{7 sin t}{8} )
C.	( frac{3 cos t}{8} )
D.	( frac{7 sin t}{9} )
Answer» C. ( frac{3 cos t}{8} )

Discussion

2.	Solve the problem of un-damped forced vibrations of a spring in the case where the forcing function is f(t)=A sin t. D.E associated with the problem is (m frac{d^2 y}{dt^2} + ky = f(t) ), with initial conditions as y(0)=y₀ and y (0)=y₁ and assume ² = k/m, =A/m.
A.	y = y<sub>0</sub> cos u2061 t + y<sub>1</sub> sin u2061 t + ( frac{ cos t}{- ^2+ ^2} )
B.	y = y<sub>0</sub> cos u2061 t + (y<sub>1</sub>/ )sin u2061 t + ( frac{cos t}{ ^2+ ^2} )
C.	y = y<sub>0</sub> cos u2061 t + (y<sub>1</sub> )sin u2061 t + ( frac{sin t}{ ^2+ ^2} )
D.	y = y<sub>0</sub> cos u2061 t + (y<sub>1</sub>/ )sin u2061 t + ( frac{ sin t}{- ^2+ ^2} )
Answer» E.

Discussion

3.	Solve the problem of resonance damped vibration of a spring .If the governing D.E is given by (m frac{d^2 y}{dt^2} + c frac{dy}{dt} + ky=0; ) c>0 with initial conditions as y(0)=y₀ and y (0)=y₁ and assume c/m=2 , k/m= ² and (v = sqrt{ ^2- ^2} ).
A.	y = e<sup>- t</sup> (y<sub>0</sub> cos u2061vt + y<sub>1</sub> sin u2061vt)
B.	y = e^(- t) (y<sub>0</sub> cos u2061vt+ ( left( frac{y_1+ y_0}{v} right) ) sin u2061vt)
C.	y = e^(- t) ( ( left( frac{y_1+ y_0}{v} right) ) cos u2061vt + y<sub>0</sub> sin u2061vt)
D.	y = e<sup>- t</sup> (y<sub>0</sub> cos u2061vt+ y<sub>1</sub> sin u2061vt)
Answer» C. y = e^(- t) ( ( left( frac{y_1+ y_0}{v} right) ) cos u2061vt + y<sub>0</sub> sin u2061vt)

Discussion

4.	A particle moves along the x-axis according to the law ( frac{d^2 x}{dt^2} + 6 frac{dx}{dt} + 25x = 0. ) If the particle is started at x=0 with an initial velocity of 12ft/sec to the left, determine x in terms of t.
A.	x=-3e<sup>-3t</sup> sin u20614t
B.	x=-e<sup>-3t</sup> (sin u20614t+cos 4t)
C.	x=-3e<sup>-3t</sup> (sin u20614t+cos u20614t)
D.	x=-3e<sup>-3t</sup> cos u20614t
Answer» B. x=-e<sup>-3t</sup> (sin u20614t+cos 4t)

Discussion

Explore topic-wise MCQs in Ordinary Differential Equations.

A particle undergoes forced vibrations according to the law x (t) + 25x(t) = 21 sin t. If the particle starts from rest at t=0, find the displacement at any time t>0.

Solve the problem of un-damped forced vibrations of a spring in the case where the forcing function is f(t)=A sin t. D.E associated with the problem is (m frac{d^2 y}{dt^2} + ky = f(t) ), with initial conditions as y(0)=y0 and y (0)=y1 and assume 2 = k/m, =A/m.

Solve the problem of resonance damped vibration of a spring .If the governing D.E is given by (m frac{d^2 y}{dt^2} + c frac{dy}{dt} + ky=0; ) c>0 with initial conditions as y(0)=y0 and y (0)=y1 and assume c/m=2 , k/m= 2 and (v = sqrt{ ^2- ^2} ).

A particle moves along the x-axis according to the law ( frac{d^2 x}{dt^2} + 6 frac{dx}{dt} + 25x = 0. ) If the particle is started at x=0 with an initial velocity of 12ft/sec to the left, determine x in terms of t.

Solve the problem of un-damped forced vibrations of a spring in the case where the forcing function is f(t)=A sin t. D.E associated with the problem is (m frac{d^2 y}{dt^2} + ky = f(t) ), with initial conditions as y(0)=y₀ and y (0)=y₁ and assume ² = k/m, =A/m.

Solve the problem of resonance damped vibration of a spring .If the governing D.E is given by (m frac{d^2 y}{dt^2} + c frac{dy}{dt} + ky=0; ) c>0 with initial conditions as y(0)=y₀ and y (0)=y₁ and assume c/m=2 , k/m= ² and (v = sqrt{ ^2- ^2} ).