108年 - 108 國立臺北大學_學士班曁進修學士班轉學生招生考試試題_電機工程學系3年級：工程數學#147426

1.Find the particular solution of the following differential equation: (10%)
$\frac{dy}{dx} - xy = 2x, \quad y(0) = 2$

2.Find the general solution of the following differential equation: (10%)
$\frac{dy}{dx} = (x + y + 1)^2$

3.Solve (10%)
$\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 4x^2$

4.Find the general solution y(x) of the following differential equation (10%)
$y''-(y')^2=0$

5.Find the Fourier series of the saw tooth wave function
\( f(x)=x+\pi \), if -π<x<π  and  f (x+2π)=f(x) (10%)

6.Find the Fourier transform of
$f(x) = \begin{cases} a, & |x| \le a \\ 0, & |x| > a \end{cases}$
where a>0 and is a constant. (10%)

7.Find the inverse Laplace transform for (10%)
$F(s) = \ln(1 + \frac{1}{s^2})$

8.Solve the initial value problem using Laplace transform technique (10%)
$\frac{d^2y}{dt^2} - 3\frac{dy}{dt} + 2y = 4t + e^{3t}, \quad y(0) = 1, \quad \frac{dy(0)}{dt} = -1$

9.If the vectors $\vec{v}(t) = 4t\vec{i} + 5t^3\vec{k}$ and $\vec{w}(t) = t^2\vec{i} - 5t\vec{j} + 2t^3\vec{k}$. Find
$\frac{d}{dt}(\vec{w} \cdot \vec{v}) \quad (10\%)$
and
$\frac{d}{dt}(\vec{v} \times \vec{w}) \quad (10\%)$