differential equations 6.1-6.3
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DIFFERENTIAL EQUATIONS 6.1-6.3
Nolan Kwit, Austin Trinh, and Nick Amoroso!
Sample questions
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) dx
which gives
y = x 2 + x + C.
As a practice, verify that the solution obtained satisfy the differential equation given above.
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) dx
Let 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)
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 3x
Integrate both sides of the equation
ò y ' dx = ò - e 3x dx
Let 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
Exercises
Exercises: Solve the following differential equations.
a) 2y ' = 6x
b) y ' cos x = sin(2x)
c) y ' e x = e 3x
Solutions
Solutions to the above exercises
a) 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
Growth and decay
Y=Ce^kt http://www.spsu.edu/math/Dillon/225
4/SlideShows/diffeqsgandd/sld001.htm
http://www.math.ucsb.edu/~mckernan/Teaching/05-06/Winter/3C/l_7.pdf
The End
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