§2.1. Differential Equations and Solutions #3, 4, 17, 20 ...

MATH 315 PS #1

Summer 2010

?2.1. Differential Equations and Solutions

#3, 4, 17, 20, 24, 35

PS1 ?2.1#3 Show that y(t)=C e-(1/ 2)t2 is a general solution of the differential equation y = ?ty. Use a computer or calculator to sketch the solutions for the given values of the arbitrary constant C= ?3, ?2, ..., 3. Experiment with different intervals for t until you have a plot that shows what you consider to be the most important behavior of the family.

y(t) =C e-0.5t2

y(t) = ?Ct e-0.5t2 = t y(t)

y(t) =t y(t)

PS1 ?2.1#4 Show that y(t)=2t ? 2 + C e-t is a general solution of the differential equation y + y = 2t. Use a computer or calculator to sketch the solutions for the given values of the arbitrary constant C= ?3, ?2, ..., 3. Experiment with different intervals for t until you have a plot that shows what you consider to be the most important behavior of the family.

y(t) = 2t ? 2 + C e-t

y(t) = 2 ? C e-t

y(t) + y(t) = 2t

DIFFERENTIAL EQUATIONS

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MATH 315 PS #1

Summer 2010

PS1 ?2.1#17 Plot the direction field for the differential equation y = y + t by hand. Do this by drawing short lines of the appropriate slope centered at each of the integer valued coordinates (t,y), where ?2t2 and ?1y1.

Table and graph:

y =y+t

t

-2 -1 0 1 2

-1 -3 -2 -1 0 1

y 0 -2 -1 0 1 2

1 -1 0 1 2 3

PS1 ?2.1#20 Plot the direction field for the differential equation y = (t2y)/(1+y2) by hand. Do this by drawing short lines of the appropriate slope centered at each of the integer valued coordinates (t,y), where ?2t2 and ?1y1.

Table and graph:

y =(t2y)/(1+y2)

t

-2 -1 0 1 2

-1 -2 -0.5 0 -0.5 -2

y

0 0 000 0

1 2 0.5 0 0.5 2

PS1 ?2.1#24 Use a computer to draw a direction field for the given first-order differential equation y = (y+t)/(y ? t), R={(t,y): ?5t5 and ?5y5}. Use the indicated bounds for your display

window. Obtain a printout and use pencil to draw a number of possible solution trajectories on the direction field. If possible, check your solutions with a computer.

DIFFERENTIAL EQUATIONS

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MATH 315 PS #1

Summer 2010

PS1 ?2.1#35 Bacteria in a petri dish is growing according to the equation dP =0.44P, where P is dt

the mass of the accumulated bacteria (measured in milligrams) after t days. Suppose that the initial mass of the bacterial sample is 1.5 mg. Use a numerical solver to estimate the amount of bacteria after 10 days.

The initial differential equation is

P = 0.44P , P(0) = 1.5

Base on the graph, we estimate that P(10) 124. Thus, there are approximately

124 mg of bateria present after 10 days.

DIFFERENTIAL EQUATIONS

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MATH 315 PS #1

Summer 2010

?2.2. Solutions to Separable Equations

#3, 6, 10, 22

PS1 ?2.2#3 Find the general solution of the differential equation y = ex-y . If possible find an explicit solution.

dy = e x e- y dx

e y dy= e x dx

e y = e x +C

y(x)=ln( e x +C)

PS1 ?2.2#6 Find the general solution of the differential equation y = y e x ?2 e x +y?2. If possible

find an explicit solution.

dy

=

y e x ?2 e x +

y

?

2

=

e x (y-2)+(y-2)

=

( e x +1)(y-2)

dx

dy =( e x +1)dx

y-2

dy = (e x +1)dx

y-2

ln|y-2|= e x +x+C

|y-2|= e x2 +x+C

y-2=A e x2 +x

y = A ex2+x + 2

PS1 ?2.2#10 Find the general solution of the differential equation x y ?y =2x2y. If possible find an explicit solution.

dy

x =2x2y + y = y(2x2 + 1)

dx

dy 2x 2 + 1

=

dx

yx

dy =

(2x + 1 )dx

ln|y|= x 2 +ln|x|+C

y

x

e |y|= x2 +ln|x|+C

y=A xe x2

PS1 ?2.2#22 Find exact solutions for the differential equation y =(y2+1)/y, y(1)=2. State the

interval of existence. Plot each exact solution on the interval of existence. Use a numerical solver to

duplicate the solution curve for the initial value problem.

dy y 2 + 1

=

dx y

ln| y 2 +1|=2x + C

y dy =dx y2 +1 y 2 +1= e 2x+C

Initial condition y(1)=2

y(x)= ? Ae2x - 1

? Ae2 -1 =2

Ae2 -1 =2

So, the particular solution is y(x)= 5e2x-2 - 1

The interval of existence:

5e 2x-2 - 1 > 0

5e 2x-2 > 1

1

d ( y 2 + 1)

=

dx

2 y2 +1

y 2 = A e2x -1

A e 2 -1=4

A=5 e-2

e2x > e2 5

2x> 2-ln5

DIFFERENTIAL EQUATIONS

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MATH 315 PS #1

Plot of the exact solution:

ln 5

x>1-

2

x(1- ln 5 ,+ ) 2

Summer 2010

Use numerical solver to duplicate the solution curve:

DIFFERENTIAL EQUATIONS

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