108年 - 108 國立臺北大學_日間學士班轉學生招生考試試題_電機工程學系日間學士班3年級：工程數學#147488

1. (10%) Determine whether the given first-order differential equation is linear or nonlinear in the dependent variable of y.
$(y^2 - 1)dx + xdy = 0; \text{ in } y$

(1) Prove the given differential equation is exact.

(2) If it is exact, solve the given initial-value problem.

3. (10%) Solve the given differential equation of \( dy - \left(1 + e^{y - x + 5}\right)dx = 0 \)

4. (15%) Find the complementary \( (y_{c}) \) and particular solution \( (y_{p}) \) of \( y^{n} - 4y^{s} + 4y = 8x^{2} + 12 - 16e^{2s} \)

5. (15%) Evaluate the given Laplace transform of \( L\{2 + 2t^4 + e^{2t} + \sin 2t\} \)

6. (15%) Find the given inverse Laplace transform of \( L^{-1}\left\{\frac{2s - 4}{(s^2 + s)(s^2 + 1)}\right\} \)

7. (10%) Let the vector field of \( \mathbf{F} = 4xy\mathbf{i} + (2x^2 + 2yz)\mathbf{j} + (3z^2 + y^2)\mathbf{k} \), find (1) the curl and (2) the divergence.

(1) Prove the given line integral is independent of the path.

(2) Find a potential function \( \phi \)

(3) Find the given line integral.