Question

Solve ds/dt = cos ( 2t ) + sin t explicitly when t = π and s = 1.

help

Answer 1

To solve the differential equation ds/dt = cos(2t) + sin(t), we can integrate both sides with respect to t:

∫ds = ∫cos(2t) + sin(t) dt

s = 1/2 sin(2t) - cos(t) + C

where C is an arbitrary constant of integration.

To find the value of C, we can use the initial condition s = 1 when t = π:

1 = 1/2 sin(2π) - cos(π) + C

1 = 0 - (-1) + C

C = 2

Therefore, the solution to the differential equation ds/dt = cos(2t) + sin(t) with initial condition s = 1 when t = π is:

s = 1/2 sin(2t) - cos(t) + 2

Substituting t = π gives:

s = 1/2 sin(2π) - cos(π) + 2

s = 1/2 (0) - (-1) + 2

s = 3