1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Westkost [7]
3 years ago
8

Consider the first order separable equation

Mathematics
1 answer:
frutty [35]3 years ago
7 0

Answer with Step-by-step explanation:

We are given that first order separable equation

yy'=x(1+y^2)

y(0)=1

y\frac{dy}{dx}=x(1+y^2)

\frac{y}{1+y^2}dy=xdx

Taking integration on both sides

\int\frac{y}{1+y^2}dy=\int xdx

Suppose 1+y^2=u

Differentiate w.r.t u

2ydy=du

ydy=\frac{1}{2}du

Substitute the values

\frac{1}{2}\int \frac{du}{u}=\int xdx

\frac{1}{2}lnu+C=\frac{x^2}{2}

By using the formula

\int x^ndx=\frac{x^{n+1}}{n+1}+C

\int \frac{dx}{x}=ln x+C

ln(1+y^2)+2C=x^2

ln(1+y^2)+C'=x^2

x^2-ln(1+y^2)=C'

Suppose f(y)=-ln(1+y^2)

Then, the solution

x^2+f(y)=C'

Substitute t=0

0-ln(1+1)=C'

C'=-ln 2

Substitute the values

x^2=ln(1+y^2)-ln 2

x^2=ln\frac{1+y^2}{2}

By using identity ln x-ln y=ln\frac{x}{y}

\frac{1+y^2}{2}=e^{x^2}

By using identity lnx=y\implies x=e^y

1+y^2=2e^{x^2}

y^2=2e^{x^2}-1

y=\sqrt{2e^{x^2}-1}

You might be interested in
Please helpppppppppp
ankoles [38]

Answer:

wrdfrdfrDFDFDFzfsdfffq

Step-by-step explanation:

dfdfgdf d f df

6 0
3 years ago
What is 40+56 in GCF
KengaRu [80]

The GCF stands for greatest common factor. To represent a sum by its GCF we need to use the distributive property and we need to first find the GCF of the numbers. Let's break each number by its factors:

\begin{gathered} 40=2\cdot2\cdot2\cdot5 \\ 56=2\cdot2\cdot2\cdot7 \end{gathered}

We now multiply the numbers that appear on both.

\text{GCF}=2\cdot2\cdot2=8

We now apply the distributive property:

8\cdot(5+7)

3 0
2 years ago
The approximate volume of a cylindrical water tower is 7,540 ft3. The diameter of the water tower is 20 ft. What is the approxim
wlad13 [49]
I think the answer is C
8 0
3 years ago
Find the measure of b and d given that m||n.
MrRa [10]

Answer:

b = 61.1° , d = 43.7°

Step-by-step explanation:

b and 61.1° are alternate angles and are congruent , so

b = 61.1°

d and 43.7° are alternate angles and are congruent , so

d = 43.7°

5 0
2 years ago
Karen says that the equation 3( x - 3) + 5 <br> = 3x + 1 + 4
4vir4ik [10]

Answer:

8x

Step-by-step explanation:

3x-9=3x

8x

3 0
3 years ago
Other questions:
  • 20% of a number is x. What is 100% of the number. Assume x&gt;0
    8·1 answer
  • Q. A copy machine can print 405 copies in one minute. How many copies can be produced in 35 minutes
    15·1 answer
  • there are 2 aircraft carriers , A&amp;B, and carrier a is longer in length than carrier B. The total length of these 2 carriers
    11·1 answer
  • What is the value of this expression when g=-3.5
    7·1 answer
  • OK i have a problem i have to find the percentage that the price went up so i have .57 cents and then it went up to 62 cents sho
    6·1 answer
  • How any is?<br> 8m + 27 = 2m - 3 <br> with procedure and testing
    13·2 answers
  • Point slope form of (1, 0) and (6, -3)
    15·1 answer
  • Please help me i really need help please
    15·1 answer
  • BRAINIEST This is due today whoever answers this will be given brainiest
    11·1 answer
  • Do you understand?
    7·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!