Question

Find r(t) if r'(t) = t⁷ i + eᵗ j + 4te⁴t k and r(0) = i + j + k.

Answer 1

To find r(t), integrate r'(t) with respect to t and use the given initial condition r(0).

Step-by-step explanation:

The question asks to find r(t) given r'(t) = t⁷ i + eᵗ j + 4te⁴t k and r(0) = i + j + k.

To find r(t), we need to integrate r'(t) with respect to t. Integrating each component separately, we get:

• r(t) = ∫ (t⁷) dt = (1/8)t⁸ + C₁
• r(t) = ∫ (eᵗ) dt = eᵗ + C₂
• r(t) = ∫ (4te⁴t) dt = (2t²e⁴t) + C₃

Using the initial condition r(0) = i + j + k, we can find the values of the constants:

• r(0) = (1/8)(0)⁸ + C₁ = i gives us C₁ = i
• r(0) = e⁰ + C₂ = j gives us C₂ = j - 1
• r(0) = (2(0)²e⁴(0)) + C₃ = k gives us C₃ = k

Therefore, the solution is r(t) = (1/8)t⁸ + i + (eᵗ - 1)j + (2t²e⁴t)k.