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
forsale [732]
3 years ago
12

Solve the initial value problem.

Mathematics
1 answer:
antiseptic1488 [7]3 years ago
4 0

Separate the variables:

y' = \dfrac{dy}{dx} = \dfrac{2x}{1+2y} \implies (1+2y) \, dy = 2x \, dx

Integrate both sides:

\displaystyle \int(1+2y) \, dy = \int 2x \, dx[/tex\[tex]y + y^2 = x^2 + C

Use the given initial condition to solve for C :

1 + 1^2 = 1^2 + C \implies C = 1

So the particular solution is

\boxed{y + y^2 = x^2 + 1}

which you can also solve explicitly for y as a function of x. By completing the square on the left side, we have

y + y^2 = x^2 + 1

\dfrac14 + y + y^2 = x^2 + \dfrac54

\left(\dfrac12 + y\right)^2 = x^2 + \dfrac54

\dfrac12 + y = \pm \sqrt{x^2+\dfrac54}

Note that y(1) = 1 is positive, so the right side should involve the positive square root:

\dfrac12 + y = \sqrt{x^2+\dfrac54}

\boxed{y = -\dfrac12 + \sqrt{x^2+\dfrac54} = -\dfrac12 + \dfrac12 \sqrt{4x^2+5}}

You might be interested in
Write nine and eighty-four hundredths in standard form.
Scrat [10]
The answer is: 9.84
4 0
4 years ago
Read 2 more answers
What is the slope in the linear equation y = 2x=1?
Bumek [7]

Answer:

The slope of the line is 2

Step-by-step explanation:

y=2x+1, compare it to the slope intercept form of a line, y=mx+c, the slope will be 2

5 0
3 years ago
Read 2 more answers
What is the measurement of the smaller angle
kondor19780726 [428]

Answer:

IS 60 DEGRES

Step-by-step explanation:

U TIMES THE ANGLE

8 0
3 years ago
Read 2 more answers
Simplify 15x + 2y + 7x + 8y
omeli [17]

Answer:

22x+10y

Step-by-step explanation:

15x + 2y + 7x + 8y ----> 22x+10y

6 0
3 years ago
What is the mean of variable a<br><br><br><br><br><br>piz help me i have one more question
timama [110]
I think the mean of variable a is 4
4 0
3 years ago
Other questions:
  • 712,684.506 word form
    6·1 answer
  • A 26' long painting is how many inches long?
    13·1 answer
  • 1.) You need to buy grass seed for a new lawn. The lawn will cover a rectangular plot that is 60 feet by 90 feet. Each ounce of
    7·2 answers
  • Explain how to model 32% on a 10-by-10 grid. How does the model represent ratio of 32 to 100.
    6·1 answer
  • In a recent semester at a local university, 490 students enrolled in both General Chemistry and Calculus I. Of these students, 6
    13·1 answer
  • What is the slope of the line below?
    13·2 answers
  • a jogger runs 4 miles per hour faster downhill than uphill. if the jogger can run 5 miles downhill in the same time that it take
    7·2 answers
  • How do you do this ?
    6·1 answer
  • Find all solutions to the equation in the interval [0, 2pi). COS X = sin 2x​
    15·1 answer
  • Which expression is undefined?<br> ОА<br> A 0:6<br> B. 6:0<br> C-14:3<br> O 0 14 = (-3)<br> D
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!