In the following, verify by substitution that the given functions are a solution of the given differential equation (primes denote derivatives with respect to x): y" + 4y = 0; y₁ = cos 2x, y₂ = sin 2x

A radioactive material decays at a rate which is proportional to the amount of radioactivity remaining. If the amount of radioactivity remaining is denoted by N at the time t, form an equation which represents this statement. If the half-life (the time taken for half the radioactivity to decay) is 10 years, how long will it take for ¾ of the radioactivity to decay?

I need this explained in details. Can you help me learn how to solve this problem? Thank you!

The number of horsepower needed to overcome a wind drag on an automobile under design is given by P(s)=0.011s2+0.005s−0.031, where s is the speed of the car in miles per hour. At what speed will the car need to use 178 horsepower to overcome the wind drag? Round your answer to the hundredth.