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Differential Equations 6.1-6.3

Nolan Kwit , Austin Trinh, and Nick Amoroso!. Differential Equations 6.1-6.3. 6.1 Slope Fields. slope field calculator. Sample questions. 6.2 Differential Equations: Growth and Decay.

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Differential Equations 6.1-6.3

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  1. Nolan Kwit, Austin Trinh, and Nick Amoroso! Differential Equations 6.1-6.3

  2. 6.1 Slope Fields • slope field calculator

  3. Sample questions

  4. 6.2 Differential Equations: Growth and Decay • Example 1: Solve and find a general solution to the differential equation. y ' = 2x + 1 Solution to Example 1:Integrate both sides of the equation. ò y ' dx = ò (2x + 1) dxwhich gives y = x 2 + x + C. As a practice, verify that the solution obtained satisfy the differential equation given above.

  5. Examples continued… • Example 2: Solve and find a general solution to the differential equation. 2 y ' = sin(2x) Solution to Example 2:Write the differential equation of the form y ' = f(x). y ' = (1/2) sin(2x) Integrate both sides ò y ' dx = ò (1/2) sin(2x) dxLet u = 2x so that du = 2 dx, the right side becomes y = ò (1/4) sin(u) du Which gives. y = (-1/4) cos(u) = (-1/4) cos (2x)

  6. Examples… • Example 3: Solve and find a general solution to the differential equation. y 'e -x + e 2x = 0 Solution to Example 3:Multiply all terms of the equation by e x and write the differential equation of the form y ' = f(x). y ' = - e 3xIntegrate both sides of the equation ò y ' dx = ò - e 3xdxLet u = 3x so that du = 3 dx, write the right side in terms of u y = ò (-1/3) e u du Which gives. y = (-1/3) e u = (-1/3) e 3x

  7. Exercises • Exercises: Solve the following differential equations. a) 2y ' = 6x b) y ' cos x = sin(2x) c) y ' e x = e 3x

  8. Solutions • Solutions to the above exercisesa) y = (3/2) x 2 + C b) y = -2 cos x + C c) y =(1 / 2) e 2x + C • http://www.analyzemath.com/calculus/Differential_Equations/simple.html

  9. Growth and decay • Y=Ce^kt • http://www.spsu.edu/math/Dillon/2254/SlideShows/diffeqsgandd/sld001.htm • http://www.math.ucsb.edu/~mckernan/Teaching/05-06/Winter/3C/l_7.pdf

  10. The End

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