1. Calculus II (MAT 146) Dr. Day Monday, Oct 23, 2017
Differential Equations (Chapter 9)
Solutions to Differential Equations (9.2 & 9.3)
Graphical: Slope Fields (9.2 part 1)
Numerical: Euler’s Method (9.2 part 2)
Analytical: Separation of Variables (9.3)
Applications of Differential Equations
Monday, October 23, 2017
MAT 146

2. knowing that (0,0) satisfies y
Solve the differential equation graphically by generating a slope field and then sketching in a solution:
y’ = 2x – y
knowing that (0,0) satisfies y
Monday, October 23, 2017
MAT 146

3. with initial conditions (0,0)
y’ = 2x – y
with initial conditions (0,0)
Monday, October 23, 2017
MAT 146

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9. Solving Differential Equations
Solve for y: y’ = −y2
Monday, October 23, 2017
MAT 146

10. Separable Differential Equations
Monday, October 23, 2017
MAT 146

11. Separable Differential Equations
Solve for y: y’ = 3xy
Monday, October 23, 2017
MAT 146

12. Separable Differential Equations
Solve for z: dz/dx+ 5ex+z = 0
Monday, October 23, 2017
MAT 146

13. Separable Differential Equations
Monday, October 23, 2017
MAT 146

14. Separable Differential Equations
Monday, October 23, 2017
MAT 146

15. Applications!
Rate of change of a population P, with respect to time t, is proportional to the population itself.
Monday, October 23, 2017
MAT 146

16. Rate of change of the population is proportional to the population itself.
Slope Fields
Euler’s Method
Separable DEs
Monday, October 23, 2017
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.
Monday, October 23, 2017
MAT 146

18. Separable DEs
Monday, October 23, 2017
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).
Monday, October 23, 2017
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.
Monday, October 23, 2017
MAT 146