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Transcript 
Robert Mařı́k
July 14, 2006
R OBERT M A Ř ÍK
Separable ODE
file sep1.tex
First order differential equation
with separated variables
Interactive tests
Theory
Tests
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Look at three or four or twenty my quizzes and
then fill in my
please!
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Theory
Definition 1 (ODE with separated variables) Let f and g be continuous functions. The
equation
y0 = f ( x ) g(y)
(1)
is said to be an ordinary differential equation with separated variables.
Omitting constant solutions, which are solutions of g(y) = 0, we can write the equation in
the form
y0
= f (x)
g(y)
and replacing y0 =
dy
and multiplying by dx we get1
dx
1
dy = f ( x ) dx
g(y)
The words ”separate variables” mean: convert the equation into the form (2).
The general solution of (1) is
Z
1
dy =
g(y)
Z
f ( x ) dx + C,
where both integrals denote one of the primitive functions.
R OBERT M A Ř ÍK
Separable ODE
file sep1.tex
1.
Theory
Tests
(2)
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1
1
The functions f ( x ) and
are unique, up to a common constant multiple.
g(y)
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2.
Tests
yy0 = 4x
1. Find constant solution(-s). Write the number or numbers separated by commas. Write
the word empty if there si no constant solution,
R OBERT M A Ř ÍK
Separable ODE
file sep1.tex
Quiz Find the general solution of the ODE with separated variables.
2. Separate variables:
3. Integrate:
Theory
Tests
• Correct answers to the first question include ydy = 4 x dx, ydy - 4 x dx=0 and
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any constant multiple.
• Correct answers to the second question include yˆ2/2 = 2xˆ2-C, yˆ2/2 - 2xˆ2+1=C Print
and any constant multiple of this equation.
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Quiz Find the general solution of the ODE with separated variables.
1. Find constant solution(-s). Write the number or numbers separated by commas. Write
the word empty if there si no constant solution,
2. Separate variables:
R OBERT M A Ř ÍK
Separable ODE
file sep1.tex
1
1
−
y0 = 0
x+1 y−1
3. Integrate:
Theory
Tests
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Print
Quiz Find the general solution of the ODE with separated variables.
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x2 + y0 cos(y) + 1 = 0
1. Find constant solution(-s). Write the number or numbers separated by commas. Write
the word empty if there si no constant solution,
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2. Separate variables:
3. Integrate:
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Quiz Find the general solution of the ODE with separated variables.
1. Find constant solution(-s). Write the number or numbers separated by commas. Write
the word empty if there si no constant solution,
2. Separate variables:
3. Integrate:
R OBERT M A Ř ÍK
Separable ODE
file sep1.tex
y(y + y0 ) + 1 = 0
Theory
Tests
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Quiz Find the general solution of the ODE with separated variables.
x2 + 1
yy0
+ 2
=0
x
y −1
1. Find constant solution(-s). Write the number or numbers separated by commas. Write
the word empty if there si no constant solution,
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2. Separate variables:
3. Integrate:
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Quiz Find the general solution of the ODE with separated variables.
1. Find constant solution(-s). Write the number or numbers separated by commas. Write
the word empty if there si no constant solution,
2. Separate variables:
3. Integrate:
R OBERT M A Ř ÍK
Separable ODE
file sep1.tex
( x − 1) y3 − e x y 0 = 0
Theory
Tests
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Quiz Find the general solution of the ODE with separated variables.
y0 =
y−1
x 2 y2
1. Find constant solution(-s). Write the number or numbers separated by commas. Write
the word empty if there si no constant solution,
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2. Separate variables:
3. Integrate:
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Quiz Find the general solution of the ODE with separated variables.
1. Find constant solution(-s). Write the number or numbers separated by commas. Write
the word empty if there si no constant solution,
2. Separate variables:
3. Integrate:
R OBERT M A Ř ÍK
Separable ODE
file sep1.tex
2(1 + e x )yy0 = e x
Theory
Tests
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Print
Quiz Find the general solution of the ODE with separated variables.
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(y2 − 1) + yy0 ( x + 1) = 0
1. Find constant solution(-s). Write the number or numbers separated by commas. Write
the word empty if there si no constant solution,
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2. Separate variables:
3. Integrate:
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Quiz Find the general solution of the ODE with separated variables.
2x + 1
2( y − 1)
1. Find constant solution(-s). Write the number or numbers separated by commas. Write
the word empty if there si no constant solution,
2. Separate variables:
R OBERT M A Ř ÍK
Separable ODE
file sep1.tex
y0 =
3. Integrate:
Theory
Tests
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Quiz Find the general solution of the ODE with separated variables.
y0 =
2x − 1
y
x2
1. Find constant solution(-s). Write the number or numbers separated by commas. Write
the word empty if there si no constant solution,
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2. Separate variables:
3. Integrate:
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Quiz Find the general solution of the ODE with separated variables.
1. Find constant solution(-s). Write the number or numbers separated by commas. Write
the word empty if there si no constant solution,
2. Separate variables:
3. Integrate:
R OBERT M A Ř ÍK
Separable ODE
file sep1.tex
√
y0 = 2 y ln x
Theory
Tests
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Print
Quiz Find the general solution of the ODE with separated variables.
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y0 e x
2 +y
= − x/y
1. Find constant solution(-s). Write the number or numbers separated by commas. Write
the word empty if there si no constant solution,
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2. Separate variables:
3. Integrate:
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1. Find constant solution(-s). Write the number or numbers separated by commas. Write
the word empty if there si no constant solution,
2. Separate variables:
R OBERT M A Ř ÍK
Separable ODE
file sep1.tex
Quiz Find the general solution of the ODE with separated variables.
q
y0 cos2 x = (1 + cos2 x ) 1 − y2
3. Integrate:
Theory
Tests
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Quiz Find the general solution of the ODE with separated variables.
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y0 + xy = y
1. Find constant solution(-s). Write the number or numbers separated by commas. Write
the word empty if there si no constant solution,
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2. Separate variables:
3. Integrate:
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Quiz Find the general solution of the ODE with separated variables.
1. Find constant solution(-s). Write the number or numbers separated by commas. Write
the word empty if there si no constant solution,
2. Separate variables:
3. Integrate:
R OBERT M A Ř ÍK
Separable ODE
file sep1.tex
y 0 = x 2 (1 + y2 )
Theory
Tests
Quiz Find the general solution of the ODE with separated variables.
y ln y + xy0 = 0
Home Page
Print
1. Find constant solution(-s). Write the number or numbers separated by commas. Write
the word empty if there si no constant solution,
2. Separate variables:
3. Integrate:
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