1. Calculus II (MAT 146) Dr. Day Friday, March 23, 2018
Differential Equations (Chapter 9)
Numerical Representations of Solutions to Differential Equations (9.2)
Analytical Solutions to Differential Equations (9.3)
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7. Solving Differential Equations
Solve for y: y’ = −y2
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8. Solving Differential Equations
Solve for y: y’ = −y2
MAT 146

9. Solve for y: y’ = −y2, IC: (0,1)
Friday, March 23, 2018
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10. Separable Differential Equations
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11. Separable Differential Equations
Solve for y: y’ = 3xy
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12. Separable Differential Equations
Solve for z: dz/dx+ 5ex+z = 0
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13. Separable Differential Equations
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14. Separable Differential Equations
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15. Applications!
Rate of change of a population P, with respect to time t, is proportional to the population itself.
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MAT 146

16. Rate of change of the population is proportional to the population itself.
Slope Fields
Euler’s Method
Separable DEs
Friday, March 23, 2018
MAT 146

17. Population Growth
Suppose a population increases by 3% each year and that there are P=100 organisms initially present (at t=0). Write a differential equation to describe this population growth and then solve for P.
Friday, March 23, 2018
MAT 146

18. Separable DEs
Friday, March 23, 2018
MAT 146

19. Applications!
The radioactive isotope Carbon-14 exhibits exponential decay. That is, the rate of change of the amount present (A) with respect to time (t) is proportional to the amount present (A).
Friday, March 23, 2018
MAT 146

20. Exponential Decay
The radioactive isotope Carbon-14 exhibits exponential decay. That is, the rate of change of the amount present (C) with respect to time (t) is proportional to the amount present (C).
Carbon-14 has a half-life of 5730 years
Write and solve a differential equation to determine the function C(t) to represent the amount, C, of carbon-14 present, with respect to time (t in years), if we know that 20 grams were present initially. Use C(t) to determine the amount present after 250 years.
Friday, March 23, 2018
MAT 146

21. Applications!
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MAT 146

22. Applications!
The security staff at a rock concert found a dead body in a mezzanine restroom, the apparent victim of a fatal shooting. They alert the police who arrive at precisely 12 midnight. At that instant, the body’s temperature is 91º F; by 1:30 a.m., 90 minutes later, the body’s temperature has dropped to 82º F. Noting that the thermostat in the restroom was set to maintain a constant temperature of 69º F, and assuming the the victim’s temperature was 98.6º F when she was shot, determine the time, to the nearest minute, that the fatal shooting occurred. Assume that the victim died instantly and that Newton’s Law of Cooling holds. Show all appropriate evidence to support your solution.
Friday, March 23, 2018
MAT 146

23. Applications!
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24. Friday, March 23, 2018
MAT 146