39. Consider the following spring-oscillation problem - the second order linear differential equation
m
d
2
dx
2
y + ky = A cos(αt) + B sin(αt)
where the positive constant m > 0 represents mass, the positive constant k > 0 represents
spring constant, the external force is given by f(t) = A cos(αt) + B sin(αt). Given that
resonance occurred in the system. Which of the following statements is true?
(A) y(t) = C1 cos r
k
m
t
!
+ C2 sin r
k
m
t
!
+
1
2
r
m
k
Bt cos r
k
m
t
!
+
1
2
r
m
k
Atsin r
k
m
t
!
(B) y(t) = C1 cos r
k
m
t
!
+ C2 sin r
k
m
t
!
−
1
2
r
m
k
Bt cos r
k
m
t
!
+
1
2
r
m
k
Atsin r
k
m
t
!
(C) y(t) = C1 cos r
k
m
t
!
+ C2 sin r
k
m
t
!
−
1
2
r
k
m
At cos r
k
m
t
!
+
1
2
r
k
m
Btsin r
k
m
t
!
(D) y(t) = C1 cos r
k
m
t
!
+ C2 sin r
k
m
t
!
+
1
2
r
k
m
At cos r
k
m
t
!
+
1
2
r
k
m
Btsin r
k
m
t
!

The given differential equation for the spring-oscillation problem is:

\[
m \frac{d^2 y}{dt^2} + ky = A \cos(\alpha t) + B \sin(\alpha t)
\]

This is a second-order linear non-homogeneous differential equation, where \( m > 0 \) represents the mass, \( k > 0 \) represents the spring constant, and the external force is given by \( f(t) = A \cos(\alpha t) + B \sin(\alpha t) \).

### Understanding Resonance:

Resonance occurs when the frequency of the external force matches the natural frequency of the system. The natural frequency of the spring-mass system is given by:

\[
\omega_0 = \sqrt{\frac{k}{m}}
\]

For resonance to occur, the frequency of the external force must match this natural frequency. The external force has components \( \cos(\alpha t) \) and \( \sin(\alpha t) \) with frequency \( \alpha \). Resonance occurs when:

\[
\alpha = \omega_0 = \sqrt{\frac{k}{m}}
\]

### Solving the Differential Equation:

The complementary solution to the homogeneous part \( m \frac{d^2 y}{dt^2} + ky = 0 \) is:

\[
y_c(t) = C_1 \cos\left(\sqrt{\frac{k}{m}} t\right) + C_2 \sin\left(\sqrt{\frac{k}{m}} t\right)
\]

For the particular solution, since resonance is occurring (\( \alpha = \sqrt{\frac{k}{m}} \)), the standard approach leads to solutions involving a term proportional to \( t \) (since the standard sinusoidal response is not sufficient in the resonant case).

The particular solution will be of the form:

\[
y_p(t) = t \left( A' \cos\left(\sqrt{\frac{k}{m}} t\right) + B' \sin\left(\sqrt{\frac{k}{m}} t\right) \right)
\]

Substituting this into the differential equation and matching coefficients with the external force term, we get:

\[
y_p(t) = \frac{1}{2} \sqrt{\frac{m}{k}} \left( A t \sin\left(\sqrt{\frac{k}{m}} t\right) - B t \cos\left(\sqrt{\frac{k}{m}} t\right)

Explore the spring-oscillation problem with resonance in differential equations. Learn how to solve and understand the behavior of spring-mass systems under external forces. Suitable for advanced undergraduate students and those interested in mechanical vibrations.

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