Solution Manual For Differential Equations And Linear Algebra, 3rd Edition

Solution Manual For Differential Equations And Linear Algebra, 3rd Edition gives you all the tools you need to solve your textbook problems effectively.

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1Solutions to Section 1.1True-False Review:1.FALSE.A derivative must involvesomederivative of the functiony=f(x), not necessarily the firstderivative.2.TRUE.The initial conditions accompanying a differential equation consist of the values ofy, y′, . . .att= 0.3. TRUE.If we define positive velocity to be oriented downward, thendvdt=g,wheregis the acceleration due to gravity.4.TRUE.We can justify this mathematically by starting froma(t) =g, and integrating twice to getv(t) =gt+c, and thens(t) = 12gt2+ct+d, which is a quadratic equation.5. FALSE.The restoring force is directed in the directionoppositeto the displacement from the equilibriumposition.6.TRUE.According to Newton’s Law of Cooling, the rate of cooling is proportional to thedifferencebetween the object’s temperature and the medium’s temperature.Since that difference is greater for theobject at 100◦Fthan the object at 90◦F, the object whose temperature is 100◦Fhas a greater rate ofcooling.7.FALSE.The temperature of the object is given byT(t) =Tm+ce−kt, whereTmis the temperatureof the medium, andcandkare constants.Sincee−kt6= 0, we see thatT(t)6=Tmfor all timest.Thetemperature of the objectapproachesthe temperature of the surrounding medium, but never equals it.8. TRUE.Since the temperature of the coffee is falling, the temperaturedifferencebetween the coffee andthe room is higher initially, during the first hour, than it is later, when the temperature of the coffee hasalready decreased.9. FALSE.The slopes of the two curves arenegativereciprocals of each other.10. TRUE.If the original family of parallel lines have slopeskfork6= 0, then the family of orthogonal tra-jectories are parallel lines with slope−1k. If the original family of parallel lines are vertical (resp. horizontal),then the family of orthogonal trajectories are horizontal (resp. vertical) parallel lines.11. FALSE.The family of orthogonal trajectories for a family of circles centered at the origin is the familyof lines passing through the origin.Problems:1.Starting from the differential equationd2ydt2=g, wheregis the acceleration of gravity andyis the unknownposition function, we integrate twice to obtain the general equations for the velocity and the position of theobject:dydt=gt+c1andy(t) =gt22+c1t+c2,wherec1, c2are constants of integration. Now we impose the initial conditions:y(0) = 0 implies thatc2= 0,

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Differential Equations And Linear A...

Solution Manual For Differential Equations And Linear Algebra, 3rd Edition

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