Chapter 6.1, Problem 2E

Show that the function $y(t)=e^{\sin t}$ is a solution of the initial value problems $\text{(a) }y\text{'}=y\cos t,y(0)\text{1 (b) }y"=(\cos t)y\text{'}-(\sin t)y,y(0)=1,y\text{'}(0)=\text{1}$ $\text{(c) }y"=y\left(1-\ln y-{\left(\ln y\right)}^2\right),y(\pi )=1,y(\pi )=-1$