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

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

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

Monday, October 23, 2017 MAT 146

Monday, October 23, 2017 MAT 146

Monday, October 23, 2017 MAT 146

Monday, October 23, 2017 MAT 146

Monday, October 23, 2017 MAT 146

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

Separable Differential Equations Monday, October 23, 2017 MAT 146

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

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

Separable Differential Equations Monday, October 23, 2017 MAT 146

Separable Differential Equations Monday, October 23, 2017 MAT 146

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

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

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

Separable DEs Monday, October 23, 2017 MAT 146

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

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