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Chapter 8: Q8-3-19P (page 404)
If in (3.10), then. Findfor this case.
The value of the given expression is, .
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By using Laplace transforms, solve the following differential equations subject to the given initial conditions.
A solution containing 90% by volume of alcohol (in water) runs at 1 gal/min into a 100-gal tank of pure water where it is continually mixed. The mixture is withdrawn at the rate of 1 gal/min. When will it start coming out 50% alcohol?
Find the orthogonal trajectories of each of the following families of curves. In each case, sketch or computer plot several of the given curves and several of their orthogonal trajectories. Be careful to eliminate the constant from for the original curves; this constant takes different values for different curves of the original family, and you want an expression for which is valid for all curves of the family crossed by the orthogonal trajectory you are trying to find. See equations to
Suppose the rate at which bacteria in a culture grow is proportional to the number present at any time. Write and solve the differential equation for the number N of bacteria as a function of time t if there are bacteria when . Again note that (except for a change of sign) this is the same differential equation and solution as in the preceding problems.
Find the solutions of (1.2)and (1.3), if ( const.).
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