Initial Value Problems, Slope Fields Section 6.1a

Determine which graph shows the solution of the given initial value problem without actually solving the problem. (–1,1) The correct graph: Can you explain why this is the correct graph??? Do Now: #26 on p.313

An equation like containing a derivative is a differential equation. Initial Value Problem – the problem of finding a function y of x when given its derivative and its value at a particular point. Initial Condition – the value of f for one value of x. Solution to the Differential Equation – all of the functions y that satisfy the differential equation. Solution to the Initial Value Problem – a particular solution that fulfills the initial condition.

An equation like containing a derivative is a differential equation. Differential equations from previous chapters??? IIIImplicit differentiation yields differential equations RRRRelated rates equations are differential equations

First New Example Suppose $100 is invested in an account that pays 5.6% interest compounded continuously. Find a formula for the amount in the account at any time t. Let t = 0 when the initial $100 is deposited in the account We can model this situation with the initial value problem:

First New Example Suppose $100 is invested in an account that pays 5.6% interest compounded continuously. Find a formula for the amount in the account at any time t. We need a function whose derivative is a constant multiple of itself…  Exponential functions have this property! …because:

First New Example Suppose $100 is invested in an account that pays 5.6% interest compounded continuously. Find a formula for the amount in the account at any time t. Now, apply the initial condition: In this case,

First New Example Suppose $100 is invested in an account that pays 5.6% interest compounded continuously. Find a formula for the amount in the account at any time t. In general, each member of the family of functions is a solution of the differential equation Let’s learn how to “see” this family of functions…

Definition: Slope Field (Direction Field) A slope field or direction field for the first order differential equation is a plot of short line segments with slopes for a lattice of points in the plane. They are useful for “seeing” solutions to differential equations even when explicit solutions are difficult to come by…

Guided Practice Plot the solution curves of the differential equation Let’s use a new calculator program to see the slope field, as well as some specific solution curves… Can we find the original function???

Guided Practice Solve the given initial value problem. Support your answer by overlaying your solution on a slope field for the differential equation. Solution: Slope Field and Graph? Initial Condition:

Guided Practice Solve the given initial value problem. Solution:

Guided Practice Solve the given initial value problem. Solution:

Guided Practice Solve the given initial value problem. First Derivative:

Guided Practice Solve the given initial value problem. Solution:

Guided Practice Use the given information about a body to find the body’s position s at time t.