To find the solution r(t) of the differential equation, we need to integrate each component of r'(t) with respect to t.
For the first component, integrating sin(5t) with respect to t gives us -(1/5)cos(5t) + C1, where C1 is the constant of integration.
For the second component, integrating sin(8t) with respect to t gives us -(1/8)cos(8t) + C2, where C2 is the constant of integration.
For the third component, integrating 3t with respect to t gives us (3/2)t^2 + C3, where C3 is the constant of integration.
Therefore, the solution r(t) is given by:
r(t) = ⟨-(1/5)cos(5t) + C1, -(1/8)cos(8t) + C2, (3/2)t^2 + C3⟩.
To find the specific solution r(t) with the initial condition r(0) = ⟨8, 2, 9⟩, we substitute t = 0 into the solution and solve for the constants C1, C2, and C3.
r(0) = ⟨-(1/5)cos(5*0) + C1, -(1/8)cos(8*0) + C2, (3/2)*0^2 + C3⟩
⟨8, 2, 9⟩ = ⟨-(1/5) + C1, -1/8 + C2, C3⟩.
By comparing the corresponding components, we get the equations:
-(1/5) + C1 = 8
-1/8 + C2 = 2
C3 = 9
Solving these equations, we find:
C1 = 8 + 1/5 = 41/5
C2 = 2 + 1/8 = 17/8
C3 = 9
Therefore, the specific solution r(t) with the initial condition r(0) = ⟨8, 2, 9⟩ is:
r(t) = ⟨-(1/5)cos(5t) + 41/5, -(1/8)cos(8t) + 17/8, (3/2)t^2 + 9⟩.
Find the solution r(t)
of the differential equation with the given initial condition:
r′(t)=⟨sin5t,sin8t,3t⟩,r(0)=⟨8,2,9⟩
1 answer