Answer:
[tex]u=-x^4-2x^2y+y-y^2[/tex]
Step-by-step explanation:
We are given that
[tex](1-2x^2-2y)\frac{dy}{dx}=4x^3+4xy[/tex]
We have to solve the given differential equation
[tex](1-2x^2-2y)dy=(4x^3+4xy)dx[/tex]
[tex](1-2x^2-2y)dy-(4x^3+4xy)dx=0[/tex]
Compare with [tex]Mdx+ndy=0[/tex]
Then, we get [tex]M=-(4x^3+4xy),N=(1-2x^2-2y)[/tex]
Exact differential equation
[tex]M_y=N_x[/tex]
[tex]M_y=-4x[/tex]
[tex]N_x=-4x[/tex]
[tex]M_y=N_x[/tex]
Hence, the differential equation is an exact differential equation.
Solution of exact differential is given by
[tex]u=\int M(x,y)dx+K(y)[/tex] where K(y) is a function of y.
[tex]u=\int -(4x^3+4xy) dx+k(y)[/tex] y treated as constant
[tex]u=-x^4-2x^2y+k(y)[/tex]
[tex]u_y=N[/tex]
[tex]-2x^2+k'(y)=1-2x^2-2y[/tex]
[tex]K'(y)=1-2y[/tex]
[tex]k(y)=y-y^2[/tex]
Substitute the value then we get
Then, [tex]u=-x^4-2x^2y+y-y^2[/tex]
Final answer:
The given differential equation is exact, and by integrating the respective terms and finding the function H(y), the solution for z(x, y) for the differential equation is determined to be
-x⁴ + y - y² + C, where C is a constant.
Explanation:
To solve the differential equation (1-2x² - 2y)dy/dx = 4x³ + 4xy, let's rearrange the equation in the form M(x, y)dy + N(x, y)dx = 0, which is the standard form for a first-order differential equation. We can rewrite our equation as (-4x³ - 4xy)dx + (1-2x² - 2y)dy = 0.
Now, we check if the equation is exact, that is, if ∂M/∂y = ∂N/∂x. If it is exact, there exists a function z(x, y) such that dz = M dy + N dx.
In this case, ∂M/∂y = -4x, and ∂N/∂x = -4x. Since the partial derivatives are equal, the differential is exact, meaning there exists some function z(x, y) such that dz = Ndx + Mdy. To find z(x, y), we integrate N with respect to x and M with respect to y and combine the resulting functions, making sure to include the functions of the other variable that may arise from the partial integration.
For N(x, y), we integrate -4x³dx to get -x⁴. For M(y), we integrate (1-2y)dy to get y - y². Thus z(x, y) = -x⁴ + y - y² + H(y), where H(y) is a function of y.
To find H(y), differentiate z with respect to y and equate it to M: dz/dy = 1 - 2y + H'(y) = 1 - 2y, which implies that H'(y) = 0, hence H(y) is a constant. Therefore, the function that satisfies the differential equation is z(x, y) = -x⁴ + y - y² + C, where C is a constant.
How do I compete the two-way table for #4
Factor: 4x^2-12x+9=0
Johann maintains a$75,000 insurance policy on his boat. He pays $50 per month in premium and his deductible is $2,500 if he is involved in a accident and make a claim of 15,000 on the policy, how much will his insurance company pay ?
A. $2,500
B.$12,500
C.$15,000
D.$17,500
Answer:
B) 12,500
Step-by-step explanation:
Just took the quiz :)
The sum of the angle measures of any triangle is
180
degrees. Suppose that one angle in a triangle has a degree measure of
7x+3
and another has a degree measure of
3x-6
. Write an expression for the degree measure of the third angle in the triangle.
Let f(x) = -4x + 7 and g(x) = 2x - 6. Find (f.g)(1)
Answer:
The answer would be 23
Step-by-step explanation:
I think this is a composite function, so it would be
f(g(1))
you must first replace all the "x" in f(x) with g(x)
f(g(x) = -4(g(x))+7
f(g(x)) = -4(2x-6)+7
f(g(x)) = -8x+24+7
f(g(x)) = -8x+31
If x=1
f(g(1)) = -8*1+31
f(g(1)) = -8+31
f(g(1)) = 23
Find the half-life of an element which decays by 3.401% each day?
The half life of an element which decays 3.401% is 20 days.
What is Half Life?The time required for a radioactive substance (or one-half of its atoms) to decay or change into a new substance is commonly characterised as its half-life. Ernest Rutherford discovered the idea for the first time in 1907. It is commonly denoted by the symbols Ug or t1/2.
To help you comprehend the notion, consider a radioactive element with a half-life of one hour. In this example, one half would decay in an hour and the other half would decay in another hour.
We are given;
y = a[tex]b^{(t)}[/tex]
Where t is the half life
y = a/2 is the amount of substance remaining after decay
So, b = 100% - 3.401%
b = 96.599%
b = 0.96599
Now, a/2 = a[tex](0.96599)^t[/tex]
and, 0.5= [tex](0.96599)^t[/tex]
Now, taking log on both side we get
ln (0.5) = t(ln 0.96599)
t = ln(0.5)/ln(0.96589)
t = 20.032 days
Learn more about Half life here:
https://brainly.com/question/4389576
#SPJ3
What is 4.75 as an improper fraction in simplest form
The solution is : 4.75 as an improper fraction in simplest form is 19/4.
Here, we have,
given that,
4.75 as an improper fraction
we need to find the simplest form
we have,
4.75
= 4 75/100
= 475/100
= ( we can divide numerator and denominator by 25 )
=475/25 /100/ 25
= 19/4
[ in simplest form]
Answer:
4.75 as an improper fraction in simplest form is 19/4.
To learn more on division click:
brainly.com/question/21416852
#SPJ6
One angle is twice another angle. The sum of the angles is 90 degrees. How many degrees are in each angle. Please explain.
18 + x > 40
a. x > 58
b. x > 22
c. x > -58
d. x > -22
How do I find the 25th term of the arithmetic sequence -39, -31, -23, -15,...
Which is the only fraction that must be represented by a non terminating decimal?
a. 7/5
b. 7/4
c. 7/3
d. 7/2
Answer:
The correct option is C) 7/3.
Step-by-step explanation:
Consider the provided numbers.
If the rational number can be written in the form of [tex]\frac{p}{2^n\times 5^m}[/tex] then the rational number will be a terminating decimal.
Now consider the provided options:
Option A) 7/5 can be written as: [tex]\frac{7}{2^0\times 5^1}[/tex]
Thus it is a terminating decimal.
Option B) 7/4 can be written as: [tex]\frac{7}{2^2\times 5^0}[/tex]
Thus it is a terminating decimal.
Option C) 7/3 can be written as: [tex]\frac{7}{2^0\times 5^0\times 3^1}[/tex]
Thus it is not a terminating decimal.
Option D) 7/2 can be written as: [tex]\frac{7}{2^1\times 5^0}[/tex]
Thus it is a terminating decimal.
Hence the only fraction that must be represented by a non terminating decimal is 7/3.
The correct option is C) 7/3.
Roland Corporation buys stoves from a wholesaler. The list price of a stove is $900, with a trade discount of 30 percent. The net price is: ...?
100% - 30% = 70%
P = 0.7*900
which equals $630
Answer:
The net price of the stove is $630.
Step-by-step explanation:
The list price of a stove = $900
Trade discount on the price = 30%
The net price = 900 - (30% of 900)
= 900 - ([tex]\frac{30}{100}[/tex] × 900)
= 900 - (0.30 × 900)
= 900 - 270
= $630
The net price of the stove is $630.
The formula for the volume of a cube is V(s) = s³ where s is the side length of the cube. What is the domain and range of this function?
s < 0, V(s) < 0
s > 0, V(s) < 0
s < 0, V(s) > 0
s > 0, V(s) > 0
What is 3x-5y=12 with a substitution of X=3y
What is the next number in the sequence?
1,4,13,40,121,...
A sector of a circle is shown. Find its area, rounded to the nearest tenth.
a)9.8 cm^2
b)10.5 cm^2
c)11.2 cm^2
d)12.3 cm^2 ...?
The area of the sector of the circle whose radius is 4.2 cm while the angle made at the center is 68° is 10.5 cm².
What is a circle?A circle can be defined as a shape that consists of all points in a plane that are at equal distance from a given point, therefore, the center.
As it is given that the length of the radius of the circle is 4.2 cm, while the angle made by the sector at the center of the circle is 68°. Therefore, the area of the sector of the circle can be written as,
[tex]\text{Area of the sector of the circle} = \pi r^2\times \dfrac{\theta}{360^o}[/tex]
Substituting the value, we will get,
[tex]\text{Area of the sector of the circle} = \pi \times(4.2^2)\times \dfrac{68^o}{360^o}[/tex]
[tex]= 10.472 \approx 10.5\rm\ cm^2[/tex]
Hence, the area of the sector of the circle whose radius is 4.2 cm while the angle made at the center is 68° is 10.5 cm².
Learn more about Circle:
https://brainly.com/question/11833983
Triangle ABC is similar to triangle DEF. The scale factor from triangle ABC to triangle DEF is 1/3. If DE = 6 m, find the length of AB.
1
18
2
3
If a fish weighs 8 times as much as her parakeet if they both weigh 63 ounces together how much does the fish weigh alone
Loren puts £600 in a bank account.
The account pays 3% compound interest each year after ONE year she withdraws £200.
How much will she have in the account after 2 years?
To solve the problem we will first calculate the remaining balance in Loren's Account after a year.
The balance in Loren's account after two years is £430.54.
Given to us
Loren puts £600 in a bank account,the account pays 3% compound interest each year,after ONE year Loren withdraws £200Principal Amount that Loren put in her account, P = £600Rate of interest that bank pays, r = 3% = 0.03Balance of Loren after one yearUsing the compound interest formula,
Balance of Loren after one year = [tex]P(1+r)^t[/tex]
Substituting values,
[tex]=600(1+0.03)^1\\= 600(1.03)\\= 618[/tex]
Remaining Balance of Loren after one yearNow, as Loren takes £200 out of the account, the remaining balance would be,
= Balance of Loren after one year - £200
= £618 - £200
= £418
Balance in Loren's account after two yearsIn the second year, the balance will be paid on the remaining balance, therefore, applying the formula of compound interest,
P = £418
r = 0.03
t = 1
[tex]=P(1+r)^t\\=418(1+0.03)^1\\=418(1.03)\\= 430.54[/tex]
Hence, the balance in Loren's account after two years is £430.54.
Learn more about Compound Interest:
https://brainly.com/question/25857212
What is the value of y? 3(y 5)=–2y3(y 5)=–2y
a. –5
b. –5
c. –3
d. –335?
Does a translation produce congruent figures?
Given f(x) = the quantity of 4x plus 1, divided by 3, solve for f−1(3).
2
4
6
8
Answer:
2
Step-by-step explanation:
Here, the given function is,
f(x) = The quantity of 4x plus 1, divided by 3
[tex]\implies f(x)=\frac{4x+1}{3}[/tex]
The steps that involve in finding the inverse of f(x) are as follows,
Step 1 : Replace f(x) by y,
[tex]y=\frac{4x+1}{3}[/tex]
Step 2 : Switch x and y,
[tex]x=\frac{4y+1}{3}[/tex]
Step 3 : Isolate y in the left side of the equation,
[tex]3x=4y+1[/tex]
[tex]-4y=1-3x[/tex]
[tex]y=\frac{1-3x}{-4}[/tex]
[tex]y=\frac{3x-1}{4}[/tex]
Step 4 : Replace y by [tex]f^{-1}(x)[/tex],
[tex]f^{-1}(x)=\frac{3x-1}{4}[/tex]
For x = 3,
[tex]f^{-1}(3)=\frac{3\times 3-1}{4}=\frac{9-1}{4}=\frac{8}{4}=2[/tex]
Hence, First option is correct.
Marcie wants to enclose her yard with a fence. Her yard is in the shape of a rectangle attached to a triangle. The formula for the area of the enclosed space is A = lw + 0.5bh. Solve for b.
A. b = A divided by the quantity l times w plus 0.5 times h
B. b = The quantity A minus l times w all divided by 0.5 times h
C. b = A – lw – 0.5h
D. b = A + lw + 0.5h
No Idea :o
Convert 2/5 to a percent
2/5 to a percent would be 40%.
The work is attached in the image provided.
For which values of x and y is line p parallel to line q?
X=5,y=3
x=1, y=5
x=3, y=5
x=3, y=6
Step 1
Find the value of x
we know that
If lines p and q are parallel
then
[tex](45x-5)+(16x+2)=180\°[/tex] ------> by consecutive interior angles
Solve for x
[tex](45x-5)+(16x+2)=180\° \\61x=180+3 \\x=3\°[/tex]
Step 2
Find the value of y
we know that
If lines p and q are parallel
then
[tex](26y)=(45x-5)[/tex] ------> by corresponding angles
substitute the value of x and solve for y
[tex](26y)=(45*3-5)[/tex]
[tex](26y)=130[/tex]
[tex]y=5\°[/tex]
therefore
the answer is the option
[tex]x=3\°[/tex] , [tex]y=5\°[/tex]
A father is 1 year older than 3 times the age of his son. The son is 20 years old.
Let F = the age of the father.
Which formula would calculate the father's age?
1. F=3(20-1)
2. F + 1 = 3(20)
3. F - 1 = 3(20)
6 2/3 greater than 6 12/15
Ariane gets paid $0.50 for the first 100 craft kits that she assembles. She earns $0.75 for each kit over 100. This week she assembled 135 kits. How much did she earn?
A. $67.50
B. $72.50
C. $76.25
D. $101.25
Final answer:
Ariane earned $50 for the first 100 kits at $0.50 per kit, and for the additional 35 kits at $0.75 per kit, she earned $26.25. Adding these amounts, her total earnings are $76.25.
Explanation:
To calculate Ariane's earnings for assembling 135 craft kits, we need to consider two different pay rates: one rate for the first 100 kits and a higher rate for each kit above 100.
For the first 100 kits, she earns $0.50 per kit, which is 100 kits × $0.50/kit = $50. For the remaining 35 kits (135 total - 100), she earns a higher rate of $0.75 per kit, which is 35 kits × $0.75/kit = $26.25.
Now, we add both amounts to find the total earnings:
$50 (first 100 kits) + $26.25 (kits over 100) = $76.25 total earnings
Thus, Ariane earned $76.25 for assembling 135 craft kits this week.
Frank had $25 in his checking account he withdrew $15 cash and $9 to pay his bill what is the expression
Please help me . factor. X^2-13x+40
Factoring x2-13x+40
The first term is, x2 its coefficient is 1 .
The middle term is, -13x its coefficient is -13 .
The last term, "the constant", is +40
Step-1 : Multiply the coefficient of the first term by the constant 1 • 40 = 40
Step-2 : Find two factors of 40 whose sum equals the coefficient of the middle term, which is -13 .
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -8 and -5
x2 - 8x - 5x - 40
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-8)
Add up the last 2 terms, pulling out common factors :
5 • (x-8)
Step-5 : Add up the four terms of step 4 :
(x-5) • (x-8)
Which is the desired factorization