Section 6 1 euler s method
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Section 6.1: Euler’s Method. Local Linearity and Differential Equations.

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Section 6.1: Euler’s Method

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Section 6 1 euler s method

Section 6.1: Euler’s Method


Local linearity and differential equations

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 equations1

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 equations2

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 equations3

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 (2.6,1.584):

“Tangent line” at (2.6,1.584):

“Tangent line” approximation at x=2.8:


Local linearity and differential equations4

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 (2.8,2.421):

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

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

Example

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

Euler’s Method

5


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