Chapter 5: Applications of Integration
Section 5.6: Differential Equations
An algebraic equation is an open statement that is true when the "openings" in the statement are filled with the appropriate algebraic expressions. Thus, 2 x+3=5 becomes true when x, the "opening," is replaced with the number 1.
A differential equation is an open statement in which the openings are are a function and at least one of its derivatives. Table 5.6.1 lists several examples of differential equations.
Immediately integrable to the general solution yx=x+c
Knowledge of the exponential function suggests the general solution yx=c ex, but this equation is actually separable.
The equation is first-order, linear, and yields to the "recipe" y=e∫2 ⅆx∫x e∫2 ⅆx ⅆx+c.
The general first-order, linear, equation is px y′+qx y=rx.
The equation is homogeneous, and becomes first-order, linear, under the change of variables vx=yx/x.
The equation is nonlinear. If functions of either y or y′ appear in the equation, it is no longer linear.
This is a Bernoulli equation, and becomes first-order, linear, under the change of variables yx=zkx for some special value of k.
This is the general separable equation whose solution is given implicitly by direct integration: ∫fy ⅆy=∫gx ⅆx+c.
Table 5.6.1 Examples of differential equations
The separable differential equation affords great opportunity to practice evaluating both indefinite and definite integrals. It will be the main focus of this chapter.
Since the general solution of the first-order differential equation fy,y′,x=0 contains one arbitrary constant of integration, the solution of such an equation represents a family of curves. One unique member of this family can be distinguished by imposing one algebraic condition of the form yx0=y0, called an initial condition. A differential equation and its associated initial condition(s) is called an initial value problem (IVP).
An implicit solution of the IVP y′x fy=gx,yx0=y0, is ∫y0yfs ⅆs=∫x0xgs ⅆs. The differential equation is separable; and the variable of integration on each side is immaterial, as long as it does not duplicate one of the limits.
Obtain the general solution of the differential equation y′x=x yx.
Obtain the general solution to the differential equation x dx−2 y 1+x2 dy=0.
Solve the initial-value problem consisting of the differential equation x−2 y 1+x2⋅y′=0 and the initial condition y1=2. Graph the solution.
Graph the solution of the initial-value problem consisting of the differential equation y′=4⁢x⁢y2+8⁢y2+x+2, and the initial condition y1=0.
A tank contains 50 lbs of salt dissolved in 1000 gallons of water. Brine containing 1/100 lb of salt per gal of water enters the tank at a rate of 40 gal per minute, mixes instantaneously, and drains at the same rate. Determine the tank's salt content 15 minutes later.
If y>0, solve the initial-value problem consisting of the differential equation y′t=k yt and the initial condition y0=y0.
Solve the initial-value problem consisting of the logistic differential equation y./y=k c−y, and the initial condition y0=y0.
A species undergoes logistic growth, governed by the formula developed in Example 5.6.7. Observation yields the following three data points.
[Time in yearsPopulation Size113003187042070]
Determine the carrying capacity c, the initial population y0, and the rate constant k, if it is known that k>0.
Obtain the general solution of the differential equation u.=k u−us, where ut represents the temperature of a body in thermal contact with its surroundings at a fixed temperature us. This equation, sometimes called Newton's law of cooling, simply states that the rate of change of temperature of the body is proportional to the difference in temperature between the body and its surroundings.
A medical examiner (M.E.) notes the temperature of the body of a deceased person is 85°F, and the environment in which the body has been located is 65°F. Being careful not to alter the surrounding temperature, the M.E. waits 15 minutes and again checks the body's temperature, finding it to be 80°F. Using Newton's law of cooling, what estimate can the M.E. make for the time of death of the deceased?
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