Exercise

Find the general solution of the following differential equations.
$$\begin{array}{ll} (a) \ y^{\prime}\cos{x} - y\sin{x} + e^{x} ... ...{x} & (d) \ xy^{\prime} + (1 + x)y = e^{-x}\sin{2x} \end{array}$$
2. Solve the following initial problems.
$(a) \ y^{\prime} + (\cos{x})y = e^{-\sin{x}}, \ y(0) = 2$
$(b) \ (x\log{x})y^{\prime} - y = \log{x}, \ y(e) = -1$
$(c) \ y^{\prime} + y = f(x), \ y(0) = 0, \ f(x) = \left\{\begin{array}{ll} 1, & 0 \leq x < 1\\ 0, & x \geq 1 \end{array} \right.$
$(d) \ y^{\prime} + (\tan{x})y = \cos^{2}{x}, \ y(0) = -1$
3. Find $i(t)$, where $R = 10\Omega, E = 20V, L = \left\{\begin{array}{rl} 5-t,& 0 \leq t\leq 5\\ 0,& 5 \leq t \end{array} \right.$ closed cicuit with $i(0) = 0$.