Part 2

1. Find the gerneral solution of $\frac{dy}{dx}+\sin x +y \cos x= \cos x+ y\sin x$.

   Solution: $y=1+Ce^{-\sin x - \cos x}$

2. Find the general solution of $\frac{dy}{dx}+ \frac{1}{x} y = 1$

   Solution: $\frac{1}{2} x+\frac{C}{x}$

3. Find the solution to $y^{\prime \prime}-8y^{\prime}+17y=0$ that satisfy the conditions $y(0)=1$, $y^{\prime}(0)=-1$.

   Solution: $y=e^{4x} ( \cos x -5 \sin x )$

4. Suppose it takes 6 days for 25% of a radioactive substance to decay.
   1. What is the half-life of the substance? Solution: $\frac{6\ln 2}{2\ln 2-\ln 3}$
   2. How long does it take for 75% of the substance to decay? Solution: $\frac{12\ln 2}{2\ln 2-\ln 3}$