Find the solution r(t)
of the differential equation with the given initial condition:
r′(t)=⟨sin5t,sin8t,3t⟩,r(0)=⟨8,2,9⟩
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⟩.
To find the solution r(t) of the given differential equation, we need to integrate each component of the vector equation.
Let's integrate the first component:
∫r'(t)₁ dt = ∫sin(5t) dt
Using the formula for the integral of sine function, we get:
= -cos(5t)/5 + C₁
Next, let's integrate the second component:
∫r'(t)₂ dt = ∫sin(8t) dt
Using the formula for the integral of sine function, we get:
= -cos(8t)/8 + C₂
Finally, let's integrate the third component:
∫r'(t)₃ dt = ∫3t dt
Using the power rule for integration, we get:
= (3/2)t^2 + C₃
Now, we can find the values of C₁, C₂, and C₃ using the initial condition r(0) = ⟨8, 2, 9⟩.
Substituting t = 0 into the integrated components, we get:
-8/5 + C₁ = 8 (equation 1)
-1/2 + C₂ = 2 (equation 2)
3/2 * 0^2 + C₃ = 9 (equation 3)
From equation 3, we can see that C₃ = 9.
From equation 2, we can see that C₂ = 2 + 1/2 = 5/2.
From equation 1, we can see that C₁ = 8 + 8/5 = 48/5.
Now, the solution r(t) is given by the vector:
r(t) = ⟨-cos(5t)/5 + 48/5, -cos(8t)/8 + 5/2, (3/2)t^2 + 9⟩