The displacement s (in metres) of a body in a damped mechanical system, with no
external forces satisfies the following differential equation:
6 3 2
2
dt
ds
dt
d s
where t represents time in seconds. If at time t = 0, s = 0 and 5
dt
ds
m/s,
solve the differential equation for s in terms of t.
To solve the given differential equation for s in terms of t, we can use the method of separation of variables.
Step 1: Rewrite the given differential equation in standard form.
The given differential equation is:
6(d^2s/dt^2) + 3(ds/dt) + 2s = 0
Step 2: Rearrange the terms to isolate the second derivative of s:
6(d^2s/dt^2) = -3(ds/dt) - 2s
Step 3: Divide both sides of the equation by 6 to obtain:
(d^2s/dt^2) = (-1/2)(ds/dt) - (1/3)s
Step 4: Now, we can separate the variables by multiplying both sides of the equation by dt^2/ds:
dt^2 = (-1/2)(ds/dt)dt - (1/3)s(ds/dt)dt
Step 5: Integrate both sides of the equation with respect to t:
∫ dt^2 = ∫ (-1/2)(ds/dt)dt - ∫ (1/3)s(ds/dt)dt
Step 6: Evaluate the integrals on the right-hand side:
t^3/3 + C1 = (-1/2)∫ (ds/dt)dt - (1/3)∫ s(ds/dt)dt
Step 7: Integrate the first term on the right-hand side with respect to t:
t^3/3 + C1 = (-1/2)∫ ds - (1/3)∫ s(ds)
Step 8: Evaluate the integrals on the right-hand side:
t^3/3 + C1 = (-1/2)s - (1/3)(s^2/2) + C2
Step 9: Combine the constants of integration:
t^3/3 + C1 = (-1/2)s - (1/6)s^2 + C2
Step 10: Combine the constants of integration into a single constant:
t^3/3 + C1 = (-1/2)s - (1/6)s^2 + C
Step 11: Rearrange the equation to solve for s:
s^2/6 + s/2 - t^3/3 = C - C1
Step 12: Introduce a new constant, C3 = C - C1:
s^2/6 + s/2 - t^3/3 = C3
Therefore, the solution to the given differential equation for s in terms of t is:
s^2/6 + s/2 - t^3/3 = C3, where C3 is a constant.