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VashaNatasha [74]
3 years ago
14

Solve the following differential equations or initial value problems. In part (a), leave your answer in implicit form. For parts

(b) and (C), write your answer in explicit form.
a. y'= t²+7/y⁴-4y³
b. y'= (cos²y) ln t
c. (t²+t)y' +y² = ty²+, y(1)= -1
Mathematics
1 answer:
shepuryov [24]3 years ago
8 0

Answer:

(a) (y^5)/5 + y^4 = (t^3)/3 + 7t + C

(b) y = arctan(t(lnt - 1) + C)

(c) y = -1/ln|0.09(t + 1)²/t|

Step-by-step explanation:

(a) dy/dt = (t^2 + 7)/(y^4 - 4y^3)

Separate the variables

(y^4 - 4y^3)dy = (t^2 + 7)dt

Integrate both sides

(y^5)/5 + y^4 = (t^3)/3 + 7t + C

(b) dy/dt = (cos²y)lnt

Separate the variables

dy/cos²y = lnt dt

Integrate both sides

tany = t(lnt - 1) + C

y = arctan(t(lnt - 1) + C)

(c) (t² + t) dy/dt + y² = ty², y(1) = -1

(t² + t) dy/dt = ty² - y²

(t² + t) dy/dt = y²(t - 1)

(t² + t)/(t - 1)dy/dt = y²

Separating the variables

(t - 1)dt/(t² + t) = dy/y²

tdt/(t² + t) - dt/(t² + t) = dy/y²

dt/(t + 1) - dt/(t(t + 1)) = dy/y²

dt/(t + 1) - dt/t + dt/(t + 1) = dy/y²

Integrate both sides

ln(t + 1) - lnt + ln(t + 1) + lnC = -1/y

2ln(t + 1) - lnt + lnC = -1/y

ln|C(t + 1)²/t| = -1/y

y = -1/ln|C(t + 1)²/t|

Apply y(1) = -1

-1 = ln|C(1 + 1)²/1|

-1 = ln(4C)

4C = e^(-1)

C = (1/4)e^(-1) ≈ 0.09

y = -1/ln|0.09(t + 1)²/t|

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How to answer <br> 100x^2+60x+9
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Step-by-step explanation:

Rewrite

100

x

2

100

x

2

as

(

10

x

)

2

(

10

x

)

2

.

(

10

x

)

2

+

60

x

+

9

(

10

x

)

2

+

60

x

+

9

Rewrite

9

9

as

3

2

3

2

.

(

10

x

)

2

+

60

x

+

3

2

(

10

x

)

2

+

60

x

+

3

2

Check the middle term by multiplying

2

a

b

2

a

b

and compare this result with the middle term in the original expression.

2

a

b

=

2

⋅

(

10

x

)

⋅

3

2

a

b

=

2

⋅

(

10

x

)

⋅

3

Simplify.

2

a

b

=

60

x

2

a

b

=

60

x

Factor using the perfect square trinomial rule

a

2

+

2

a

b

+

b

2

=

(

a

+

b

)

2

a

2

+

2

a

b

+

b

2

=

(

a

+

b

)

2

, where

a

=

10

x

a

=

10

x

and

b

=

3

b

=

3

.

(

10

x

+

3

)

2

3 0
3 years ago
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