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
lyudmila [28]
2 years ago
13

1. (a) Solve the differential equation (x + 1)Dy/dx= xy, = given that y = 2 when x = 0. (b) Find the area between the two curves

y = x2 and y = 2x – x2.​

Mathematics
1 answer:
erastova [34]2 years ago
7 0

(a) The differential equation is separable, so we separate the variables and integrate:

(x+1)\dfrac{dy}{dx} = xy \implies \dfrac{dy}y = \dfrac x{x+1} \, dx = \left(1-\dfrac1{x+1}\right) \, dx

\displaystyle \frac{dy}y = \int \left(1-\frac1{x+1}\right) \, dx

\ln|y| = x - \ln|x+1| + C

When x = 0, we have y = 2, so we solve for the constant C :

\ln|2| = 0 - \ln|0 + 1| + C \implies C = \ln(2)

Then the particular solution to the DE is

\ln|y| = x - \ln|x+1| + \ln(2)

We can go on to solve explicitly for y in terms of x :

e^{\ln|y|} = e^{x - \ln|x+1| + \ln(2)} \implies \boxed{y = \dfrac{2e^x}{x+1}}

(b) The curves y = x² and y = 2x - x² intersect for

x^2 = 2x - x^2 \implies 2x^2 - 2x = 2x (x - 1) = 0 \implies x = 0 \text{ or } x = 1

and the bounded region is the set

\left\{(x,y) ~:~ 0 \le x \le 1 \text{ and } x^2 \le y \le 2x - x^2\right\}

The area of this region is

\displaystyle \int_0^1 ((2x-x^2)-x^2) \, dx = 2 \int_0^1 (x-x^2) \, dx = 2 \left(\frac{x^2}2 - \frac{x^3}3\right)\bigg|_0^1 = 2\left(\frac12 - \frac13\right) = \boxed{\frac13}

You might be interested in
PLZ HELP ME !!!!!!!!!!!!!!<br> 2 questions
disa [49]

Answer: D

Step-by-step explanation: I think

6 0
2 years ago
Fernando puts 3 pictures on each page of his photo album. He puts pictures on 8 pages. Represent it by array.
lakkis [162]
X X X X X X X X
X X X X X X X X
X X X X X X X X
6 0
3 years ago
Read 2 more answers
If the measure of an angle is p is three times less than twice the measure of angle q and angle p and angle q are supplementary
Marizza181 [45]
60 * 3 = 180 and 90 * 2 = 180
if they are supplementary angles than they equal 180 together. 
angle p is three times less than twice the measure of angle q so that means 180/3 = angle p and 180/2 = angle q. 
Angle q is 90 and twice that is 180 and Angle p is 60 and three times that is 180
so 60 is three times less than twice the measure of 90.
60 three times is 180 and 90 twice is 180 which means 3p = 2q and you have 180 because they are supplementary angles so you have really 180/3 = p and 180/2 = q
The answer is p = 60 and q = 90
8 0
3 years ago
Read 2 more answers
Ruth has a savings account at a bank that charges a $3.50 fee for every month her balance falls below $1,500. Her account has $1
borishaifa [10]

Answer:

1304.5 balance after 5 months

Step-by-step explanation:

first you subtract 400 from 1722 to find the balance she will have which is 1322. then you must multiply 3.5 x 5= 17.50. now subtract 17.50 from 1322 and you get her balance after 5 months

7 0
3 years ago
List the &lt; from the largest to smallest for ABC
devlian [24]

Answer: B

Step-by-step explanation:

Since AB < AC < BC, it follows that \angle C < \angle B < \angle A

4 0
2 years ago
Other questions:
  • Help me match these formulas pleasee:(
    10·1 answer
  • Find the mean of the data set \{4.5, 5, 3.5, 2, 2.5\}{4.5,5,3.5,2,2.5}left brace, 4, point, 5, comma, 5, comma, 3, point, 5, com
    13·1 answer
  • Help help help help help help help help
    7·2 answers
  • Zx and Zy are supplementary angles. Zy measures 116º.<br> What is the measure of Z?
    11·1 answer
  • Is the function y=6/x-6 linear or non linear
    6·2 answers
  • a system of equations has infinitely many solutions. if 2y – 4x = 6 is one of the equations, which could be the other equation?
    13·2 answers
  • Maria realiza una encuesta sobre las edades de los postulantes a una universidad y con los datos obtenidos elabora una tabla de
    15·1 answer
  • Any awnsers will give brainless
    7·1 answer
  • Several million lottery tickets are sold, and 21% of the tickets are held by men. Suppose you select a random sample of people a
    14·1 answer
  • Larry flips a coin and picks one out of three marbles (red, blue and yellow). How many total outcomes are possible?
    13·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!