 1.3.1: Under the same assumptions underlying the model in ( 1 ), determine...
 1.3.2: The population model given in (1) fails to take death into consider...
 1.3.3: Using the concept of a net rate introduced in 2, determine a differ...
 1.3.4: Modify the model in for the net rate at which the population P(t) o...
 1.3.5: A cup of coffee cools according to Newton's law of cooling (3). Use...
 1.3.6: The ambient temperature Tm in (3) could be a function of time t. Su...
 1.3.7: Suppose a student carrying a flu virus returns to an isolated colle...
 1.3.8: At a time t = 0, a technological innovation is introduced into a co...
 1.3.9: Suppose that a large mixing tank initially holds 300 gallons of wat...
 1.3.10: Suppose that a large mixing tank initially holds 300 gallons of wat...
 1.3.11: What is the differential equation in 10, if the wellstirred soluti...
 1.3.12: Generalize the model given in (8) on page 21 by assuming that the l...
 1.3.13: Suppose water is leaking from a tank through a circular hole of are...
 1.3.14: The rightcircular conical tank shown in FIGURE 1.3.13 loses water ...
 1.3.15: A series circuit contains a resistor and an inductor as shown in FI...
 1.3.16: A series circuit contains a resistor and a capacitor as shown in FI...
 1.3.17: For highspeed motion through the airsuch as the skydiver shown in...
 1.3.18: A cylindrical barrels ft in diameter of weight w lb is floating in ...
 1.3.19: After a mass m is attached to a spring, it stretches s units and th...
 1.3.20: In 19, what is a differential equation for the displacement x(t) if...
 1.3.21: Consider a singlestage rocket that is launched vertically upward a...
 1.3.22: In 21, suppose m(t) = mp+ mv + m1(t) where mp is constant mass of t...
 1.3.23: By Newton's law of universal gravitation, the freefall acceleratio...
 1.3.24: Suppose a hole is drilled through the center of the Earth and a bow...
 1.3.25: Learning Theory In the theory of learning, the rate at which a subj...
 1.3.26: Forgetfulness In 25, assume that the rate at which material is forg...
 1.3.27: Infusion of a Drug A drug is infused into a patient's bloodstream a...
 1.3.28: Tractrix A motorboat starts at the origin and moves in the directio...
 1.3.29: Reflecting Surface Assume that when the plane curve C shown in FIGU...
 1.3.30: Reread in Exercises 1.1 and then give an explicit solution P(t) for...
 1.3.31: Reread the sentence following equation (3) and assume that Tm is a ...
 1.3.32: Reread the discussion leading up to equation (8). lf we assume that...
 1.3.33: Population Model ThedifferentialequationdP/dt= (kcost)P where k is ...
 1.3.34: Rotating Fluid As shown in FIGURE 1.3.23(a), a rightcircular cylin...
 1.3.35: Falling Body In 23, supposer = R + s, wheres is the distance from t...
 1.3.36: Raindrops Keep Falling In meteorology, the term virga refers to fal...
 1.3.37: Let It Snow The "snowplow problem" is a classic and appears in many...
 1.3.38: Reread this section and classify each mathematical model as linear ...
 1.3.39: Population Dynamics Suppose that P' (t) = 0.15 P(t) represents a ma...
 1.3.40: Radioactive Decay Suppose that A'(t) = 0.0004332A(t) represents a ...
Solutions for Chapter 1.3: Differential Equations as Mathematical Models
Full solutions for Advanced Engineering Mathematics  5th Edition
ISBN: 9781449691721
Solutions for Chapter 1.3: Differential Equations as Mathematical Models
Get Full SolutionsThis textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 5. Since 40 problems in chapter 1.3: Differential Equations as Mathematical Models have been answered, more than 33018 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781449691721. Chapter 1.3: Differential Equations as Mathematical Models includes 40 full stepbystep solutions.

Alias
In a fractional factorial experiment when certain factor effects cannot be estimated uniquely, they are said to be aliased.

Bayes’ estimator
An estimator for a parameter obtained from a Bayesian method that uses a prior distribution for the parameter along with the conditional distribution of the data given the parameter to obtain the posterior distribution of the parameter. The estimator is obtained from the posterior distribution.

Biased estimator
Unbiased estimator.

Categorical data
Data consisting of counts or observations that can be classiied into categories. The categories may be descriptive.

Causal variable
When y fx = ( ) and y is considered to be caused by x, x is sometimes called a causal variable

Central limit theorem
The simplest form of the central limit theorem states that the sum of n independently distributed random variables will tend to be normally distributed as n becomes large. It is a necessary and suficient condition that none of the variances of the individual random variables are large in comparison to their sum. There are more general forms of the central theorem that allow ininite variances and correlated random variables, and there is a multivariate version of the theorem.

Conditional probability density function
The probability density function of the conditional probability distribution of a continuous random variable.

Continuous uniform random variable
A continuous random variable with range of a inite interval and a constant probability density function.

Control chart
A graphical display used to monitor a process. It usually consists of a horizontal center line corresponding to the incontrol value of the parameter that is being monitored and lower and upper control limits. The control limits are determined by statistical criteria and are not arbitrary, nor are they related to speciication limits. If sample points fall within the control limits, the process is said to be incontrol, or free from assignable causes. Points beyond the control limits indicate an outofcontrol process; that is, assignable causes are likely present. This signals the need to ind and remove the assignable causes.

Correlation
In the most general usage, a measure of the interdependence among data. The concept may include more than two variables. The term is most commonly used in a narrow sense to express the relationship between quantitative variables or ranks.

Covariance matrix
A square matrix that contains the variances and covariances among a set of random variables, say, X1 , X X 2 k , , … . The main diagonal elements of the matrix are the variances of the random variables and the offdiagonal elements are the covariances between Xi and Xj . Also called the variancecovariance matrix. When the random variables are standardized to have unit variances, the covariance matrix becomes the correlation matrix.

Defect
Used in statistical quality control, a defect is a particular type of nonconformance to speciications or requirements. Sometimes defects are classiied into types, such as appearance defects and functional defects.

Deining relation
A subset of effects in a fractional factorial design that deine the aliases in the design.

Error propagation
An analysis of how the variance of the random variable that represents that output of a system depends on the variances of the inputs. A formula exists when the output is a linear function of the inputs and the formula is simpliied if the inputs are assumed to be independent.

Error variance
The variance of an error term or component in a model.

Extra sum of squares method
A method used in regression analysis to conduct a hypothesis test for the additional contribution of one or more variables to a model.

Fixed factor (or fixed effect).
In analysis of variance, a factor or effect is considered ixed if all the levels of interest for that factor are included in the experiment. Conclusions are then valid about this set of levels only, although when the factor is quantitative, it is customary to it a model to the data for interpolating between these levels.

Gamma function
A function used in the probability density function of a gamma random variable that can be considered to extend factorials

Goodness of fit
In general, the agreement of a set of observed values and a set of theoretical values that depend on some hypothesis. The term is often used in itting a theoretical distribution to a set of observations.

Harmonic mean
The harmonic mean of a set of data values is the reciprocal of the arithmetic mean of the reciprocals of the data values; that is, h n x i n i = ? ? ? ? ? = ? ? 1 1 1 1 g .