1SeparationofVariables - Drexel University

Math 122 S

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1 Separation of Variables

Solve the following di ferential equations.

1.1

dy = dx

xy - y

dy = xy - y dx

dy dx

= y(

x-

)

y dy = ( x - ) dx

dy = ( x - ) dx y

ln |y| = x - x + C

y = ex -x+C

= ex -xeC

y = Cex -x

Calculus II

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1.2

dy = xy - y dx x - x

dy = xy - y dx x - x

dy dx

=

y( x

x- -x

)

y

dy

=

x- x -x

dx

We need to use partial fraction decomposition for the right side:

x- = x- x - x x(x - )

=

A x

+

x

B -

A(x - ) + Bx = x -

Letting x = , we get A = . Letting x = , we get B = .

y dy = x + x - dx

ln |y| = ln |x| + ln |x - | + C

y = eln |x|+ ln |x- |+C

= eln |x|e ln |x- |eC

y = Cx(x - )

Calculus II

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Calculus II

1.3 Word Problem

Suppose that a ball is moving along a straight line through a resistive medium in such a way that its velocity v = v(t) decreases at a rate that is twice the square root of the velocity. Suppose that at time t = seconds, the velocity of the ball is meters/second.

(a) Set up and solve an initial value problem whose solution is v = v(t). Express your solution as an explicit function of t.

(b) At what time does the ball come to a complete stop?

dv dt

=

-

v

dv = - dt v

dv = - dt v v=- t+C v = -t + C = -( ) + C

C=

v(t) = (-t + ) The ball comes to a complete stop when v = :

(-t + ) = -t + = t=

The ball comes to a complete stop after 7 seconds.

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2 Improper Integrals

2.1

dx

x-

We can start by making a substitution. Let u = x - . Then du = dx.

dx = du

x-

u

= lim du

t + t u

= lim u

t + t = lim - t

t +

=?-?

=

Calculus II

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Calculus II

2.2

xex dx

-

We can rst nd the anti derivative using integration by parts and then worry about the bounds. Let u = x and let dv = ex dx. Then du = dx and v = ex

xex dx = xex - ex dx

= xex - ex

xex dx

-

lim

t- t

xex dx = lim xex - ex

t-

t

= lim ( - ) - (tet - et)

t-

=-

Here we use the fact that lim tet = , which requires L'Hospital's rule:

t-

lim tet

t-

=

lim

t-

t e-t

=

lim

t-

-e-t

= lim -et

t-

=

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