By Peter V O'Neil
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Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 4 Homogeneous, Bernoulli, and Riccati Equations 29 This is linear when P(x) is identically zero. If we can somehow obtain one particular solution S(x) of a Riccati equation, then the change of variables y = S(x) + 1 z transforms the Riccati equation to a linear equation in x and z.
Write it as 1 dm = k dt. m A routine integration yields ln |m| = kt + c. Since mass is positive, |m| = m and m(t) = ekt+c = Aekt in which A can be any positive number. Any radioactive element has its mass decrease according to a rule of this form, and this reveals an important characteristic of radioactive decay. Suppose at some time τ there are M grams. Look for h so that, at the later time τ + h, exactly half of this mass has radiated away. This would mean that m(τ + h) = M = Aek(τ +h) = Aekτ ekh .
How much salt is in the tank when it contains 100 gallons of brine? Hint: The amount of brine in the tank at time t is 50 + t. 5. y − 2y = −8x 2 In each of Problems 6 through 10, solve the initial value problem. 6. y + 3y = 5e2x − 6; y(0) = 2 7. y + 1 x−2 y = 3x; y(3) = 4 8. y − y = 2e4x ; y(0) = −3 9. y + 10. y + 2 x+1 5y 9x y = 3; y(0) = 5 = 3x 3 + x; y(−1) = 4 11. Find all functions with the property that the y intercept of the tangent to the graph at (x, y) is 2x 2 . 12. A 500 gallon tank initially contains 50 gallons of brine solution in which 28 pounds of salt have been 5 gal/min; 1/2 lb/gal Tank 1 13.
Advanced engineering mathematics by Peter V O'Neil