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Section 6.1: Euler’s Method. Local Linearity and Differential Equations.

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Local Linearity and Differential Equations

Local Linearity can be used to numerically approximate an equation without knowing the particular solution. Consider the differential equation if . (The blue curve below is the particular solution. Assume this is unknown.)

Not a good approximation. Consider smaller pieces.

Slope at (2,0):

Tangent line at (2,0):

Tangent line approximation at x=2.2:

Local Linearity and Differential Equations

Local Linearity can be used to numerically approximate an equation without knowing the particular solution. Consider the differential equation if . (The blue curve below is the particular solution. Assume this is unknown.)

Slope at (2.2,0.4):

“Tangent line” at (2.2,0.4):

“Tangent line” approximation at x=2.4:

Local Linearity and Differential Equations

Local Linearity can be used to numerically approximate an equation without knowing the particular solution. Consider the differential equation if . (The blue curve below is the particular solution. Assume this is unknown.)

Slope at (2.4,0.92):

“Tangent line” at (2.4,0.92):

“Tangent line” approximation at x=2.6:

Local Linearity and Differential EquationsLocal Linearity can be used to numerically approximate an equation without knowing the particular solution. Consider the differential equation if . (The blue curve below is the particular solution. Assume this is unknown.)

Slope at (2.6,1.584):

“Tangent line” at (2.6,1.584):

“Tangent line” approximation at x=2.8:

Local Linearity and Differential EquationsLocal Linearity can be used to numerically approximate an equation without knowing the particular solution. Consider the differential equation if . (The blue curve below is the particular solution. Assume this is unknown.)

Slope at (2.8,2.421):

We can generalize this process.

“Tangent line” at (2.8,2.421):

“Tangent line” approximation at x=3:

Derivative

Previous y

Change in x

Next y

Euler’s Method

Numerically approximate values for the solution of the initial-value problem , , with step size , at , are

Change in x

Next Approximate Solution

Previous Approximate Solution

Value of Differential Equation at Previous Point

Example

If and if when , use Euler’s Method with five equal steps to approximate when .

Euler’s Method

5

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