EXERCISE SET 8.2 Graphing Uility CAS 1-10 Solve Where reasonable, express the fa

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EXERCISE SET 8.2 Graphing Uility CAS 1-10 Solve Where reasonable, express the family of solutions as explicitO 25. Suppose that the initial condition in hoht oe of the solui Example I satisfy this initial condition, and initial-value problem the differential equation by separation of variables. functions of x. Why does the method of Example I fail to pd ticular solution? tion in Example l isreplaced by yao of the resalting initial-value problem is defined iera 26. Find all ordered pairs Co, yo) such that if the in 27. Find an equation of a curve with a-interope 2 gent line at any point (xoy) has slope xe 11-14 Solve the initial-value problem by separation of vari 28. Use a graphing utility to generate a curve through the point (1,1) and whose targem Ei? sendiular to theline throh y+cos y 2y/(3x. 29. Suppose that an initial population of 10,000 hacte exponentially at a rate of 2% per hour and y the namber of bacteria present r hours later (a) Find an initial-value problem whose soluation is (b) Find a formula for yr). (c) How long does it take for the initial popalation of b I5. (a) Sketch some typical integral curves of the differential equation yy/2x (b) Find an equation for the integral curve that passes teria to double? (d) How long does it take for the population of bactea through the point (2, 1) 16. (a) Sketch some typical integral curves of the differential reach 45,000? (b) Find an equation for the integral curve that passes 30. A cell of the bacterium E. coli divides into tuo cels through the point (3,4). 20 minutes when placed in a nutrient culrure. LE 17-18 Solve the differential equation and then use a graphing utility to generate five integral curves for the equation. be the number of cells that are present / minutes afher gle cell is placed in the culture. Assume that the gos of the bacteria is approximated by an exponential gow model. (a) Find an initial-value problem whose solution is 0 (b) Find a formula for pú) (c) How many cells are present after 2 hours? (d) How long does it take for the number of cells to ra 18 (cos ylycos de 3 19-20 Solve the differential equation. I you have a CAS with implicit plosting capability, use the CAS to generate five inte gral curves for the equation. 1,000,0002 31. Radon-222 is a radioactive gas with a half-life of 3.83 ds 19. 21-24 True-False Determmine whether the statement is true or false. Explain your ans wer. This gas is a health hazard because it tends to get snppel in the basements of houses, and gest that homeowners seal their of the gas. Assume that 5.0 many health officials basements to prevent at the time it is sealed and that y) is 5.0 x 10 radon atoms are 21. Every differential equation of the form y'- ) is sepa- number of atoms peesent r days later 22. A differential equation of the form (b) Find a formula for yin (c) How many atomms will be ) llow long will it take for 90% of the criginal q an 21. If a radioactive element has a half-life of I minute, and if a container holds 32 g of the element at 1:00 PM, then the 32. Methylmencury is a toxi amount remaining at 1:05 PL will be 1 g and an increased risk of heart rological prown s0 days for the haman body to 24. If a population is gnowing exponentially, then the time it takes the pogulation to quadnuple is independent of the size 50% of a given of the population. from the sy

Explanation / Answer

Use separation of variables

1). (dy/dx ) = y/x

dy/y = dx/x now integrate both sides

ln |y| = ln|x| + ln|c|

|y| = |c.x| ...answer

3). dy/(1+y) = (1/2)(2x/?(1+x^2)).dx

ln|(1+y)| = (-1/2)ln(?(1+x^2)| + ln|c|

|(1+y)| = c.(1+x)^(-1/4)

11). (2y + cos y) dy = 3.x^2 dx

Now integrate both sides

y^2 + sin y = x^3 + c

13). (2y - 2).dy = (2t + 1).St

Integrate both sides

(y^2 - 2y) = t^2 +t + c ...answer

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