(use the
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 91E
Related questions
Question
#4 encircled red follow the lesson before
![BERNOULLI'S EQUATION
where P(x) and Q(x) are continuous functions and n is a real
number, is called Bernoulli equation.
A first order equation that can be written in the form
dy
dx
When n=0 or n=1, the equation is linear. For other values of n,
the substitution
Recall:
dy
dx
v = y¹-n
transforms the Bernoulli equation into a linear equation.
Linear
dx
dy
dx
Y
Linear type
ENGR. EDISON E. MOJICA
Soin is to convert
by
dx
N
4
1. y' + y = x³y²
+ y P(x)
+
Standard form:
Equation
2
y? dy
dx
Once done,
then the solution follows that
the linear DE
P(x) = Q(x)
General solution: y(i.f.) = Q(x)(i.f.) dx + C
where: i.f. integrating factor = e/P(x)dx
=
2
3
1+y₁ = = y²₁ x ²³
y dy t
dx
dx
Substitution
v = yl-n
+
2
sol'n: eliminate y
+ P(x)y = Q(x)yn
y P(x) = Y Q. (x)
Substitute
Y
2
7
+ y² = 1 = y ²³₁ x ³ ) y ²²
.
Bernoulli becomes Linear and
v=y
3/8 3/8
dv
dx
FIX
dk
dx
=
=
dy
dx
Gen Form
dy
3
V = y
1/2 (²4) = 2 (y^¹)
X
+yP(x) = Q(x)
1-n
Bernoulli to
Bernoulli's Equation
dy + y PG) = (^QG)
ox
+
3
OF Bernoulli
YPG) = (YOGG)
(-₁) (y -^-^) dy
dx
3
2
y dy
dx
Linear
n = 2
y
dv
dx
+ Y
V=Y
dv
21 22
dx
is this linear ?
dy
ölð ölő
i.F.
i.f
>/*x
X|<
1
-
1
4
x
4
x'y
1
+
+y P(x)
P(x)=
=
General solution:
dv
dx
- dv
dx
- 4
Q(x)
✓ P(x) = Q(x)
3
✓ =
=
- 10x + C
=
(v)(x¯ª) = S(-x³)(x ¯²) dx + c
1
√x²² = -√x²¯²dx +c
fax + c
Y
Y(i.f.) = [Q(x)(i.f.) dx + C
where: i.f. = integrating factor = P(x)dx
SP(x) dx
- S#1dx
za variables are
I
=
y dy
dx
dx
3
Q(x) = -x³
10x + C
Linear DE
1
bring back the original variable
- 1
- 4 Inx
: e
4
4
x³y₂ lnx + C x²y
= 2
and x
105x
4
2. 6y' 2y = xy4
6
² (Gdy - 2y=xY²)
6
dx
4
1 - 2 = x²
dx
↓
dy
dx + y (-1) =Y²³ (2²)
dy
dx
Y
General solution:
у
y
y
+ Y
Y
1-4
?
9
(1
"
=
you evda. S-5. @vox.dk
11
I]
2 =
=
P(x) = y^QG)
I
e
1
S(₁-4) (-1) dx
-18
alão
ex
1
√(-²)(-6) · (~8) (²) (³-X-3Jok)
+)(=
etc-osc
= =
2e*
2
+
BE
dy
dx
-1) ³x · √(1-4) (²2/²-) (@√(₁-4)(-3)**) **
↓
y²³²= ² [x² + ²]
У
2
-IN
dx
1
3
- xy + y
√(1-n) P(x) dx [(1-n) Q(x) √(1-n)P(x)dx dx
+ y P(x) = √^QG)
DOG)
3
Bernoulli's Equation
8 - 21/1/2
1/2 fx e²dx
by porto
Sudv=
du = dx
¼/12 [×e*- Se*d×]
dx
Jet U=X
= uv - Svdu
Sdu = Se*dx
√=e*
3. y' + ²√y=0
4.
dy
dx
+
X
=
x²y²
(use the
2 solutions)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56514a45-1715-4000-bef0-87ece83531e6%2F724f87c3-7537-436b-9bf0-61c9423350ec%2Fid1fy9s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:BERNOULLI'S EQUATION
where P(x) and Q(x) are continuous functions and n is a real
number, is called Bernoulli equation.
A first order equation that can be written in the form
dy
dx
When n=0 or n=1, the equation is linear. For other values of n,
the substitution
Recall:
dy
dx
v = y¹-n
transforms the Bernoulli equation into a linear equation.
Linear
dx
dy
dx
Y
Linear type
ENGR. EDISON E. MOJICA
Soin is to convert
by
dx
N
4
1. y' + y = x³y²
+ y P(x)
+
Standard form:
Equation
2
y? dy
dx
Once done,
then the solution follows that
the linear DE
P(x) = Q(x)
General solution: y(i.f.) = Q(x)(i.f.) dx + C
where: i.f. integrating factor = e/P(x)dx
=
2
3
1+y₁ = = y²₁ x ²³
y dy t
dx
dx
Substitution
v = yl-n
+
2
sol'n: eliminate y
+ P(x)y = Q(x)yn
y P(x) = Y Q. (x)
Substitute
Y
2
7
+ y² = 1 = y ²³₁ x ³ ) y ²²
.
Bernoulli becomes Linear and
v=y
3/8 3/8
dv
dx
FIX
dk
dx
=
=
dy
dx
Gen Form
dy
3
V = y
1/2 (²4) = 2 (y^¹)
X
+yP(x) = Q(x)
1-n
Bernoulli to
Bernoulli's Equation
dy + y PG) = (^QG)
ox
+
3
OF Bernoulli
YPG) = (YOGG)
(-₁) (y -^-^) dy
dx
3
2
y dy
dx
Linear
n = 2
y
dv
dx
+ Y
V=Y
dv
21 22
dx
is this linear ?
dy
ölð ölő
i.F.
i.f
>/*x
X|<
1
-
1
4
x
4
x'y
1
+
+y P(x)
P(x)=
=
General solution:
dv
dx
- dv
dx
- 4
Q(x)
✓ P(x) = Q(x)
3
✓ =
=
- 10x + C
=
(v)(x¯ª) = S(-x³)(x ¯²) dx + c
1
√x²² = -√x²¯²dx +c
fax + c
Y
Y(i.f.) = [Q(x)(i.f.) dx + C
where: i.f. = integrating factor = P(x)dx
SP(x) dx
- S#1dx
za variables are
I
=
y dy
dx
dx
3
Q(x) = -x³
10x + C
Linear DE
1
bring back the original variable
- 1
- 4 Inx
: e
4
4
x³y₂ lnx + C x²y
= 2
and x
105x
4
2. 6y' 2y = xy4
6
² (Gdy - 2y=xY²)
6
dx
4
1 - 2 = x²
dx
↓
dy
dx + y (-1) =Y²³ (2²)
dy
dx
Y
General solution:
у
y
y
+ Y
Y
1-4
?
9
(1
"
=
you evda. S-5. @vox.dk
11
I]
2 =
=
P(x) = y^QG)
I
e
1
S(₁-4) (-1) dx
-18
alão
ex
1
√(-²)(-6) · (~8) (²) (³-X-3Jok)
+)(=
etc-osc
= =
2e*
2
+
BE
dy
dx
-1) ³x · √(1-4) (²2/²-) (@√(₁-4)(-3)**) **
↓
y²³²= ² [x² + ²]
У
2
-IN
dx
1
3
- xy + y
√(1-n) P(x) dx [(1-n) Q(x) √(1-n)P(x)dx dx
+ y P(x) = √^QG)
DOG)
3
Bernoulli's Equation
8 - 21/1/2
1/2 fx e²dx
by porto
Sudv=
du = dx
¼/12 [×e*- Se*d×]
dx
Jet U=X
= uv - Svdu
Sdu = Se*dx
√=e*
3. y' + ²√y=0
4.
dy
dx
+
X
=
x²y²
(use the
2 solutions)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
