Separable IVP
-
Upload
samuel-olumoyeke -
Category
Documents
-
view
232 -
download
0
Transcript of Separable IVP
-
7/28/2019 Separable IVP
1/69
logo1
Solving Initial Value Problems An Example Double Check
An Initial Value Problem for a Separable
Differential Equation
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
-
7/28/2019 Separable IVP
2/69
logo1
Solving Initial Value Problems An Example Double Check
Approach
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
-
7/28/2019 Separable IVP
3/69
logo1
Solving Initial Value Problems An Example Double Check
ApproachThis approach will work as long as the general solution of the
differential equation can be computed.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
-
7/28/2019 Separable IVP
4/69
logo1
Solving Initial Value Problems An Example Double Check
ApproachThis approach will work as long as the general solution of the
differential equation can be computed.
1. Find the general solution of the differential equation.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
-
7/28/2019 Separable IVP
5/69
logo1
Solving Initial Value Problems An Example Double Check
ApproachThis approach will work as long as the general solution of the
differential equation can be computed.
1. Find the general solution of the differential equation.
2. Use the initial conditions to determine the value(s) of the
constant(s) in the general solution.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
-
7/28/2019 Separable IVP
6/69
logo1
Solving Initial Value Problems An Example Double Check
ApproachThis approach will work as long as the general solution of the
differential equation can be computed.
1. Find the general solution of the differential equation.
2. Use the initial conditions to determine the value(s) of the
constant(s) in the general solution.
3. Double check if the solution works.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
S l i I iti l V l P bl A E l D bl Ch k
-
7/28/2019 Separable IVP
7/69
logo1
Solving Initial Value Problems An Example Double Check
ApproachThis approach will work as long as the general solution of the
differential equation can be computed.
1. Find the general solution of the differential equation.
2. Use the initial conditions to determine the value(s) of the
constant(s) in the general solution.
3. Double check if the solution works.Thats it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
S l i I iti l V l P bl A E l D bl Ch k
-
7/28/2019 Separable IVP
8/69
logo1
Solving Initial Value Problems An Example Double Check
Solve the Initial Value Problem
y = xsin(x)y+ xsin(x)y
, y(0) = 1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
9/69
logo1
Solving Initial Value Problems An Example Double Check
Solve the Initial Value Problem
y = xsin(x)y+ xsin(x)y
, y(0) = 1.
y
= xsin(x)y+xsin(x)
y
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
10/69
logo1
Solving Initial Value Problems An Example Double Check
Solve the Initial Value Problem
y = xsin(x)y+ xsin(x)y
, y(0) = 1.
y
= xsin(x)y+xsin(x)
y
y = xsin(x)y+
1
y
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
11/69
logo1
Solving Initial Value Problems An Example Double Check
Solve the Initial Value Problem
y = xsin(x)y+ xsin(x)y
, y(0) = 1.
y = xsin(x)y+xsin(x)
y
y = xsin(x)y+
1
y
dy
dx = xsin(x)y
2 + 1
y
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
12/69
logo1
Solving Initial Value Problems An Example Double Check
Solve the Initial Value Problem
y = xsin(x)y+ xsin(x)y
, y(0) = 1.
y = xsin(x)y+xsin(x)
y
y = xsin(x)y+
1
y
dy
dx = xsin(x)y
2 + 1
y
y
y2 + 1dy = xsin(x) dx
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
13/69
logo1
g p
Solve the Initial Value Problem
y = xsin(x)y+ xsin(x)y
, y(0) = 1.
y = xsin(x)y+xsin(x)
y
y = xsin(x)y+
1
y
dy
dx = xsin(x)y
2 + 1
y
y
y2 + 1dy =
xsin(x) dx
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
14/69
logo1
g p
The Integral
y
y2 + 1dy.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
15/69
logo1
The Integral
y
y2 + 1dy.
Idea: Substitution u := y2 + 1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
16/69
logo1
The Integral
y
y2 + 1dy.
Idea: Substitution u := y2 + 1.du
dy= 2y, so dy =
du
2y.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
17/69
logo1
The Integral
y
y2 + 1dy.
Idea: Substitution u := y2 + 1.du
dy= 2y, so dy =
du
2y.
y
y2
+ 1
dy =
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
18/69
logo1
The Integral
y
y2 + 1dy.
Idea: Substitution u := y2 + 1.du
dy= 2y, so dy =
du
2y.
y
y2
+ 1
dy = y
u
du
2y
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
19/69
logo1
The Integral
y
y2 + 1dy.
Idea: Substitution u := y2 + 1.du
dy= 2y, so dy =
du
2y.
y
y2
+ 1
dy = y
u
du
2y
=1
2
1
udu
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
20/69
logo1
The Integral
y
y2 + 1dy.
Idea: Substitution u := y2 + 1.du
dy= 2y, so dy =
du
2y.
y
y2
+ 1
dy = y
u
du
2y
=1
2
1
udu
=1
2ln
|u
|+ c
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
21/69
logo1
The Integral
y
y2 + 1dy.
Idea: Substitution u := y2 + 1.du
dy= 2y, so dy =
du
2y.
y
y2
+ 1
dy = y
u
du
2y
=1
2
1
udu
=1
2ln
|u
|+ c
=1
2ln
y2 + 1+ c
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
22/69
logo1
The Integral
xsin(x) dx.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
23/69
logo1
The Integral
xsin(x) dx.
Idea: Integration by parts.
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
24/69
logo1
The Integral
xsin(x) dx.
Idea: Integration by parts.Integrate sin(x), differentiate x.
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
25/69
logo1
The Integral
xsin(x) dx.
Idea: Integration by parts.Integrate sin(x), differentiate x.
xsin(x) dx =
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
26/69
logo1
The Integral
xsin(x) dx.
Idea: Integration by parts.Integrate sin(x), differentiate x.
xsin(x) dx = x cos(x)
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
27/69
logo1
The Integral
xsin(x) dx.
Idea: Integration by parts.Integrate sin(x), differentiate x.
xsin(x) dx = x cos(x)
1 cos(x) dx
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
28/69
logo1
The Integral
xsin(x) dx.
Idea: Integration by parts.Integrate sin(x), differentiate x.
xsin(x) dx = x cos(x)
1 cos(x) dx
= xcos(x) +cos(x) dx
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
29/69
logo1
The Integral
xsin(x) dx.
Idea: Integration by parts.Integrate sin(x), differentiate x.
xsin(x) dx = x cos(x)
1 cos(x) dx
= xcos(x) +cos(x) dx= xcos(x) + sin(x) + c
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
30/69
logo1
The Integral
xsin(x) dx.
Idea: Integration by parts.Integrate sin(x), differentiate x.
xsin(x) dx = x cos(x)
1 cos(x) dx
= xcos(x) +cos(x) dx= xcos(x) + sin(x) + c= sin(x)xcos(x) + c
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
31/69
logo1
... Continuing Where We Left Off.
y
y2 + 1dy =
xsin(x) dx
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
32/69
logo1
... Continuing Where We Left Off.
y
y2 + 1dy =
xsin(x) dx
1
2ln
y2 + 1
=
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
33/69
logo1
... Continuing Where We Left Off.
y
y2 + 1dy =
xsin(x) dx
1
2ln
y2 + 1
= sin(x)xcos(x) + c
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
34/69
logo1
... Continuing Where We Left Off.
y
y2 + 1dy =
xsin(x) dx
1
2ln
y2 + 1
= sin(x)xcos(x) + c
lny2 + 1 = 2sin(x)2xcos(x) + c
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
35/69
logo1
... Continuing Where We Left Off.
y
y2 + 1dy =
xsin(x) dx
1
2ln
y2 + 1
= sin(x)xcos(x) + c
lny2 + 1 = 2sin(x)2xcos(x) + cy2 + 1 = e2sin(x)2xcos(x)+c
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
36/69
logo1
... Continuing Where We Left Off.
y
y2 + 1dy =
xsin(x) dx
1
2ln
y2 + 1
= sin(x)xcos(x) + c
lny2 + 1 = 2sin(x)2xcos(x) + cy2 + 1 = ke2sin(x)2xcos(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
37/69
logo1
... Continuing Where We Left Off.
y
y2 + 1dy =
xsin(x) dx
1
2ln
y2 + 1
= sin(x)xcos(x) + c
lny2 + 1 = 2sin(x)2xcos(x) + cy2 + 1 = ke2sin(x)2xcos(x)
y2 = ke2sin(x)2xcos(x) 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
38/69
logo1
... Continuing Where We Left Off.
y
y2 + 1dy =
xsin(x) dx
1
2ln
y2 + 1
= sin(x)xcos(x) + c
lny2 + 1 = 2sin(x)2xcos(x) + cy2 + 1 = ke2sin(x)2xcos(x)
y2 = ke2sin(x)2xcos(x) 1
y = ke2sin(x)2xcos(x) 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
39/69
logo1
Solving the Initial Value Problem
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
40/69
logo1
Solving the Initial Value Problem
1 = y(0)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
41/69
logo1
Solving the Initial Value Problem
1 = y(0)
y(x) =
ke2sin(x)2xcos(x) 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
42/69
logo1
Solving the Initial Value Problem
1 = y(0)
y(x) =
ke2sin(x)2xcos(x) 11 =
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
43/69
logo1
Solving the Initial Value Problem
1 = y(0)
y(x) =
ke2sin(x)2xcos(x) 11 = ke
2sin(0)20cos(0) 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
44/69
logo1
Solving the Initial Value Problem
1 = y(0)
y(x) =
ke2sin(x)2xcos(x) 11 = ke
2sin(0)20cos(0) 11 = ke0 1 =
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
45/69
logo1
Solving the Initial Value Problem
1 = y(0)
y(x) =
ke2sin(x)2xcos(x) 11 = ke
2sin(0)20cos(0) 11 = ke0 1 = k1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
46/69
logo1
Solving the Initial Value Problem
1 = y(0)
y(x) =
ke2sin(x)2xcos(x) 11 = ke
2sin(0)20cos(0) 11 = ke0 1 = k11 =
k1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
47/69
logo1
Solving the Initial Value Problem
1 = y(0)
y(x) =
ke2sin(x)2xcos(x) 11 =
ke2sin(0)20cos(0) 1
1 = ke0 1 = k11 =
k1
1 = k1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
48/69
logo1
Solving the Initial Value Problem
1 = y(0)
y(x) =
ke2sin(x)2xcos(x) 11 =
ke2sin(0)20cos(0) 1
1 = ke0 1 = k11 =
k1
1 = k1k = 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
-
7/28/2019 Separable IVP
49/69
logo1
Solving the Initial Value Problem
1 = y(0)
y(x) =
ke2sin(x)2xcos(x) 11 =
ke2sin(0)20cos(0) 1
1 = ke0 1 = k11 =
k1
1 = k1k = 2
y(x) =
2e2sin(x)2xcos(x) 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
S l i h I i i l V l P bl
-
7/28/2019 Separable IVP
50/69
logo1
Solving the Initial Value Problem
y(x) =
2e2sin(x)2xcos(x) 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
S l i h I i i l V l P bl
-
7/28/2019 Separable IVP
51/69
logo1
Solving the Initial Value Problem
y(x) =
2e2sin(x)2xcos(x) 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
D ( )
2 2sin( ) 2 cos( ) 1 S l h IVP
-
7/28/2019 Separable IVP
52/69
logo1
Does y(x) =
2e2sin(x)2xcos(x) 1 Solve the IVP
y = xsin(x)y+xsin(x)
y , y(0) = 1?
y(x) =
2e2sin(x)2xcos(x) 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
D ( )
2 2sin(x) 2xcos(x) 1 S l th IVP
-
7/28/2019 Separable IVP
53/69
logo1
Does y(x) =
2e2sin(x)2xcos(x) 1 Solve the IVP
y = xsin(x)y+xsin(x)
y , y(0) = 1?
y(x) =
2e2sin(x)2xcos(x) 1y(0) =
2e2sin(0)20cos(0) 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
D ( )
2 2sin(x) 2xcos(x) 1 S l th IVP
-
7/28/2019 Separable IVP
54/69
logo1
Does y(x) =
2e2sin(x)2xcos(x) 1 Solve the IVP
y = xsin(x)y+xsin(x)
y , y(0) = 1?
y(x) =
2e2sin(x)2xcos(x) 1y(0) =
2e2sin(0)20cos(0) 1 = 2e0 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
D ( )
2 2sin(x)2xcos(x) 1 S l th IVP
-
7/28/2019 Separable IVP
55/69
logo1
Does y(x) =
2e2sin(x)2xcos(x) 1 Solve the IVP
y = xsin(x)y+xsin(x)
y , y(0) = 1?
y(x) =
2e2sin(x)2xcos(x) 1
y(0) =
2e2sin(0)20cos(0) 1 = 2e0 1 = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
Does ( )
2 2sin(x)2xcos(x) 1 Solve the IVP
-
7/28/2019 Separable IVP
56/69
logo1
Does y(x) =
2e2sin(x)2xcos(x) 1 Solve the IVP
y = xsin(x)y+xsin(x)
y , y(0) = 1?
y(x) =
2e2sin(x)2xcos(x) 1
y(0) =
2e2sin(0)20cos(0) 1 = 2e0 1 = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
Does y(x)
2e2sin(x)2xcos(x) 1 Solve the IVP
-
7/28/2019 Separable IVP
57/69
logo1
Does y(x) =
2e2sin(x) 2xcos(x) 1 Solve the IVP
y = xsin(x)y+xsin(x)
y , y(0) = 1?
d
dx 2e2sin(x)2xcos(x) 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
Does y(x) =
2e2sin(x)2xcos(x) 1 Solve the IVP
-
7/28/2019 Separable IVP
58/69
logo1
Does y(x) =
2e2sin(x) 2xcos(x) 1 Solve the IVP
y = xsin(x)y+xsin(x)
y , y(0) = 1?
d
dx 2e2sin(x)2xcos(x) 1
=1
2
2e2sin(x)2xcos(x) 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
Does y(x) =
2e2sin(x)2xcos(x) 1 Solve the IVP
-
7/28/2019 Separable IVP
59/69
logo1
Does y(x) =
2e2sin(x) 2xcos(x) 1 Solve the IVP
y = xsin(x)y+xsin(x)
y , y(0) = 1?
d
dx 2e2sin(x)2xcos(x) 1
=1
2
2e2sin(x)2xcos(x) 12e2sin(x)2xcos(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
Does y(x) =
2e2sin(x)2xcos(x) 1 Solve the IVP
-
7/28/2019 Separable IVP
60/69
logo1
Does y(x) =
2e s (x) xcos(x) 1 Solve the IVP
y = xsin(x)y+xsin(x)
y , y(0) = 1?
d
dx 2e2sin(x)2xcos(x) 1
=2e2sin(x)2xcos(x)
2
2e2sin(x)2xcos(x) 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
Does y(x) =
2e2sin(x)2xcos(x) 1 Solve the IVP
-
7/28/2019 Separable IVP
61/69
logo1
Does y(x) =
2e ( ) ( ) 1 Solve the IVP
y = xsin(x)y+xsin(x)
y , y(0) = 1?
d
dx 2e2sin(x)2xcos(x) 1
=2e2sin(x)2xcos(x)
2
2e2sin(x)2xcos(x) 1
2cos(x)2cos(x) + 2xsin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
Does y(x) =
2e2sin(x)2xcos(x) 1 Solve the IVP
-
7/28/2019 Separable IVP
62/69
logo1
Does y(x) =
2e ( ) ( ) 1 Solve the IVP
y = xsin(x)y+xsin(x)
y , y(0) = 1?
d
dx 2e2sin(x)2xcos(x) 1
=2e2sin(x)2xcos(x)
2
2e2sin(x)2xcos(x) 1
2cos(x)2cos(x) + 2xsin(x)
=2e2sin(x)2xcos(x)
2
2e2sin(x)2xcos(x) 12xsin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
Does y(x) =
2e2sin(x)2xcos(x) 1 Solve the IVP
-
7/28/2019 Separable IVP
63/69
logo1
Does y(x) =
2e ( ) ( ) 1 Solve the IVP
y = xsin(x)y+xsin(x)
y , y(0) = 1?
d
dx 2e2sin(x)2xcos(x) 1
=2e2sin(x)2xcos(x)
2
2e2sin(x)2xcos(x) 1
2cos(x)2cos(x) + 2xsin(x)
=2e2sin(x)2xcos(x)
2
2e2sin(x)2xcos(x) 12xsin(x)
=2e2sin(x)2xcos(x)
2e2sin(x)2xcos(x) 1xsin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
i ( )
-
7/28/2019 Separable IVP
64/69
logo1
xsin(x)y+xsin(x)
y
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
i ( )
-
7/28/2019 Separable IVP
65/69
logo1
xsin(x)y+xsin(x)
y
= xsin(x)
2e2sin(x)2xcos(x) 1 + xsin(x)2e2sin(x)2xcos(x) 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
i ( )
-
7/28/2019 Separable IVP
66/69
logo1
xsin(x)y+xsin(x)
y
= xsin(x)
2e2sin(x)2xcos(x) 1 + xsin(x)2e2sin(x)2xcos(x) 1
=
xsin(x)2e2sin(x)2xcos(x) 1
2
+xsin(x)2e2sin(x)2xcos(x) 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
i ( )
-
7/28/2019 Separable IVP
67/69
logo1
xsin(x)y+xsin(x)
y
= xsin(x)
2e2sin(x)2xcos(x) 1 + xsin(x)2e2sin(x)2xcos(x) 1
=
xsin(x)2e2sin(x)2xcos(x) 1
2
+xsin(x)2e2sin(x)2xcos(x) 1
=xsin(x)
2e2sin(x)2xcos(x) 1 + 1
2e2sin(x)2xcos(x) 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
xsin(x)
-
7/28/2019 Separable IVP
68/69
logo1
xsin(x)y+xsin(x)
y
= xsin(x)
2e2sin(x)2xcos(x) 1 + xsin(x)2e2sin(x)2xcos(x) 1
=
xsin(x)2e2sin(x)2xcos(x) 1
2
+xsin(x)2e2sin(x)2xcos(x) 1
=xsin(x)
2e2sin(x)2xcos(x) 1 + 1
2e2sin(x)2xcos(x) 1
= xsin(x)2e2sin(x)2xcos(x)
2e2sin(x)2xcos(x) 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation
Solving Initial Value Problems An Example Double Check
xsin(x)
-
7/28/2019 Separable IVP
69/69
logo1
xsin(x)y+xsin(x)
y
= xsin(x)
2e2sin(x)2xcos(x) 1 + xsin(x)2e2sin(x)2xcos(x) 1
=
xsin(x)
2e2sin(x)2xcos(x) 1
2
+xsin(x)
2e2sin(x)2xcos(x) 1
=xsin(x)
2e2sin(x)2xcos(x) 1 + 1
2e2sin(x)2xcos(x) 1
= xsin(x)2e2sin(x)2xcos(x)
2e2sin(x)2xcos(x) 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
An Initial Value Problem for a Separable Differential Equation