It looks like the differential equation is
[tex] \left(x^2y + e^x\right) \,\mathrm dx - x^2\,\mathrm dy = 0[/tex]
Check for exactness:
[tex]\dfrac{\partial\left(x^2y+e^x\right)}{\partial y} = x^2 \\\\ \dfrac{\partial\left(-x^2\right)}{\partial x} = -2x[/tex]
As is, the DE is not exact, so let's try to find an integrating factor µ(x, y) such that
[tex] \mu\left(x^2y + e^x\right) \,\mathrm dx - \mu x^2\,\mathrm dy = 0[/tex]
*is* exact. If this modified DE is exact, then
[tex]\dfrac{\partial\left(\mu\left(x^2y+e^x\right)\right)}{\partial y} = \dfrac{\partial\left(-\mu x^2\right)}{\partial x}[/tex]
We have
[tex]\dfrac{\partial\left(\mu\left(x^2y+e^x\right)\right)}{\partial y} = \left(x^2y+e^x\right)\dfrac{\partial\mu}{\partial y} + x^2\mu \\\\ \dfrac{\partial\left(-\mu x^2\right)}{\partial x} = -x^2\dfrac{\partial\mu}{\partial x} - 2x\mu \\\\ \implies \left(x^2y+e^x\right)\dfrac{\partial\mu}{\partial y} + x^2\mu = -x^2\dfrac{\partial\mu}{\partial x} - 2x\mu[/tex]
Notice that if we let µ(x, y) = µ(x) be independent of y, then ∂µ/∂y = 0 and we can solve for µ :
[tex]x^2\mu = -x^2\dfrac{\mathrm d\mu}{\mathrm dx} - 2x\mu \\\\ (x^2+2x)\mu = -x^2\dfrac{\mathrm d\mu}{\mathrm dx} \\\\ \dfrac{\mathrm d\mu}{\mu} = -\dfrac{x^2+2x}{x^2}\,\mathrm dx \\\\ \dfrac{\mathrm d\mu}{\mu} = \left(-1-\dfrac2x\right)\,\mathrm dx \\\\ \implies \ln|\mu| = -x - 2\ln|x| \\\\ \implies \mu = e^{-x-2\ln|x|} = \dfrac{e^{-x}}{x^2}[/tex]
The modified DE,
[tex]\left(e^{-x}y + \dfrac1{x^2}\right) \,\mathrm dx - e^{-x}\,\mathrm dy = 0[/tex]
is now exact:
[tex]\dfrac{\partial\left(e^{-x}y+\frac1{x^2}\right)}{\partial y} = e^{-x} \\\\ \dfrac{\partial\left(-e^{-x}\right)}{\partial x} = e^{-x}[/tex]
So we look for a solution of the form F(x, y) = C. This solution is such that
[tex]\dfrac{\partial F}{\partial x} = e^{-x}y + \dfrac1{x^2} \\\\ \dfrac{\partial F}{\partial y} = e^{-x}[/tex]
Integrate both sides of the first condition with respect to x :
[tex]F(x,y) = -e^{-x}y - \dfrac1x + g(y)[/tex]
Differentiate both sides of this with respect to y :
[tex]\dfrac{\partial F}{\partial y} = -e^{-x}+\dfrac{\mathrm dg}{\mathrm dy} = e^{-x} \\\\ \implies \dfrac{\mathrm dg}{\mathrm dy} = 0 \implies g(y) = C[/tex]
Then the general solution to the DE is
[tex]F(x,y) = \boxed{-e^{-x}y-\dfrac1x = C}[/tex]
Suppose you write each letter of the alphabet on a different slip of paper and put the slips into a hat. What is the probability of drawing one slip of paper from the hat at random and getting a vowel?
Answer:
5/26 chance because there is only five vowels in the alphabet
Answer:
Not counting y as a vowel: 5/26 or a 19.23% chance
Step-by-step explanation:
There are a total of 26 letters in the alphabet.
There are 5 vowels (a, e, i, o, u)
Picking a vowel has a 5 in 26 chance of occurring.
Compare with >, < 12.6 and 12.3
Answer:
12.6>12.3
Step-by-step explanation:
one leg of a right triangle is 42 inches longer than the other leg, and the hypotenuse is 78 inches. find the lengths of the legs of the triangle
Answer:
use phythagoras formula of calculating right angles
a^2=b^2+c^2
42^2=b^2+78^2 (substitution + transposement)
b^2=72^2-42^2
b^2=3420 (introduce a square root both sides)
what you do on the left .you also do it on the right
b is the unknown side of the triangle
b=58.48 inched
The lengths of the legs of the right angle triangle is 30 inches and 72 inches.
What is a triangle?A triangle is a closed plane figure formed by joining three noncolinear points.
We know Pythagoras's theorem which states the sum of the squares of the smaller two sides of a right-angle triangle is equal to the square of the largest side.
Given one leg of a right-angle triangle is 42 inches longer than the other leg.
Assuming the smallest leg to be x inches.
∴ The other leg is (x + 42) inches also given hypotenuse is 78 inches.
∴ (x + 42)² + x² = 78².
x² + 84x + 1764 + x² = 6084.
2x² + 84x - 4320 = 0.
x² + 42x - 2160 = 0.
As length can not be negative value of x = 30 inches.
∴ The smaller leg is 30 inches and the larger leg is (30 + 42) = 72 inches.
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a farmer enough to last his 40 cattle for 35 days'if he bays more cattle how long can the same food last?consider the cattle finish the feed at the same rate
Answer:
this is unclear
Step-by-step explanation:
yoy miss typed
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Answer:
[tex]\sf 2(3\sqrt{8}+4\sqrt{20}+2\sqrt{24})[/tex]
[tex]\sf 2(3\sqrt{8})+2(4\sqrt{20}+2\sqrt{24})[/tex]
[tex]\sf 12 \sqrt{2}+16 \sqrt{5}+8 \sqrt{6}[/tex]
[tex]\sf 6 \cdot 8^{\frac{1}{2}}+8 \cdot 20^{\frac{1}{2}}+4 \cdot 24^{\frac{1}{2}}[/tex]
[tex]\sf 12 \cdot 2^{\frac{1}{2}}+16 \cdot 5^{\frac{1}{2}}+8 \cdot 6^{\frac{1}{2}}[/tex]
Step-by-step explanation:
Perimeter of a rectangle
Perimeter = 2(width + length)
Given information:
[tex]\textsf{width} = 3\sqrt{8}[/tex]
[tex]\textsf{length} = 4\sqrt{20}+2\sqrt{24}[/tex]
Equivalent Expression 1
Substitute the given information into the formula:
[tex]\sf Perimeter = 2(3\sqrt{8}+4\sqrt{20}+2\sqrt{24})[/tex]
Equivalent Expression 2
Using the distributive property law, this can also be written as:
[tex]\sf Perimeter = 2(3\sqrt{8})+2(4\sqrt{20}+2\sqrt{24})[/tex]
Equivalent Expression 3
Distribute the parentheses and simplify the radicals:
[tex]\begin{aligned}\sf Perimeter & = \sf 2(3\sqrt{8}+4\sqrt{20}+2\sqrt{24})\\& = \sf 6\sqrt{8}+8\sqrt{20}+4\sqrt{24}\\& = \sf 6\sqrt{4 \cdot 2}+8\sqrt{4 \cdot 5}+4\sqrt{4 \cdot 6}\\& = \sf 6\sqrt{4}\sqrt{2}+8\sqrt{4}\sqrt{5}+4\sqrt{4}\sqrt{6}\\& = \sf 6 \cdot 2 \sqrt{2}+8 \cdot 2 \sqrt{5}+4 \cdot 2 \sqrt{6}\\& = \sf 12 \sqrt{2}+16 \sqrt{5}+8 \sqrt{6}\end{aligned}[/tex]
Equivalent Expression 4
Distribute the parentheses and rewrite the square roots as [tex]\sf \sqrt{a}=a^{\frac{1}{2}}[/tex] :
[tex]\begin{aligned}\sf Perimeter & = \sf 2(3\sqrt{8}+4\sqrt{20}+2\sqrt{24})\\& = \sf 6\sqrt{8}+8\sqrt{20}+4\sqrt{24}\\ & = \sf 6 \cdot 8^{\frac{1}{2}}+8 \cdot 20^{\frac{1}{2}}+4 \cdot 24^{\frac{1}{2}}\end{aligned}[/tex]
Equivalent Expression 5
Rewrite the square roots as [tex]\sf \sqrt{a}=a^{\frac{1}{2}}[/tex] :
[tex]\sf Perimeter=12 \cdot 2^{\frac{1}{2}}+16 \cdot 5^{\frac{1}{2}}+8 \cdot 6^{\frac{1}{2}}[/tex]
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Step-by-step explanation:
[tex]a) \\ thank \: you[/tex]
Mrs. Harris treated her 1st period with donuts for averaging the highest on the midterm exam. She bought 25 donuts, By the end of the period, there were 3 donuts remaining. What is the percent decrease?
Answer:
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When I started here, there were 500 employees. Since then, we have added 35% more employees, so we now have a total of ______ employees
Answer: 675
Step-by-step explanation:
Since the starting amount of employees is 500 we can say that 100%=500.
Adding 35% more employees means that we're adding 35% to the existing 100%, which means we need to find 135% of 500.
To solve we must multiply 135% by 500. 135% as a decimal is 1.35, so we multiply 1.35(500)=675.
Therefore, they now have a total of 675 employees.
What is the product?
Answer:
product is the result of multiplication or an expression that identifies factors to be multiplied..
The product of the given expression (7x²)(2x³ + 5)(x² - 4x - 9) is,
[tex]14x^{7} - 56 x^6 - 126 x^5 + 35x^{4} - 140x^{3} - 315 x^2[/tex]
The product between expressions can be defined as the outcome or result obtained by multiplying two or more numbers, terms, or quantities together. It represents the combined value or the total when multiple factors are multiplied.
The given expression is,
(7x²)(2x³ + 5)(x² - 4x - 9)
Multiplying the first two expressions,
[tex](7 \times 2 x^{2+3} + 7\times 5x^2)(x^2 -4x -9)\\(14 x^5 + 35 x^2)(x^2 -4x -9)\\[/tex]
Now again multiplying these two expressions
[tex]14x^{5+2} - 4 \times 14 x^5 - 14 \times 9 x^5 + 35x^{2+2} - 35 \times 4x^{2+1} - 35 \times 9 x^2[/tex]
Simplifying this we get,
[tex]14x^{7} - 56 x^6 - 126 x^5 + 35x^{4} - 140x^{3} - 315 x^2[/tex]
Hence,
The product of the given expression is,
[tex]14x^{7} - 56 x^6 - 126 x^5 + 35x^{4} - 140x^{3} - 315 x^2[/tex] .
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Convert 15° to seconds.
(Remember, 1 degree = 60 minutes, 1 minute = 60 seconds)
Answer:
15 degree = 54,000 seconds
Step-by-step explanation:
15*60*60
Enter the equation of the following using line in slope-intercept form. slope = 3; y-intercept is -3
Answer:
y=3x-3
Step-by-step explanation:
slope intercept form is y=mx+b
you are given m as 3 and b as -3
so substitute
y=3x-3
Write 0.2 repeating as a fraction in simplest form.
0.2 Repeating Decimal is 2/9 as a Fraction
Answer:
2.9
Step-by-step explanation:
We let 0.2 be x.
Since x is recurring in 1 decimal places, we multiply it by 10.
10x = 2.2
Next, we subtract them.
10x - x = 2.2 - 0.2
9x = 2
Lastly, we divide both sides by 9 to get x as a fraction.
Answer: x = 2.9 or 2/9.
1. Find the value of x in the figure.
(1 point)
40°
O 40°
O 50°
O 140°
O 180°
Answer:
the value of x = 40
I hope it's helps you
Answer:
[tex]x = 40 \\ \\ [/tex]
it's vertical angel they are equal each other ( = )
Peter pay $79.80 for returning a bicycle for 6 hours, what was the rate per hour
Answer:
$13.3 per hour
Step-by-step explanation:
for 6 hour the cost is $79.80
for 1 hour the cost is $79.80/6
$13.3 per hour
Thr area of a square is 36 centimeters what is the length of each square
Answer:
6 cm
Step-by-step explanation:
The area of a square is its side squared(multiplied by itself). In order to find the length of the square, you would need to find a number which is multiplied by itself to get 36. This number is 6, meaning the length of the square is 6 cm.
The trolley rolled at a rate of 10 feet per second how far does it go in a minute
9514 1404 393
Answer:
600 feet
Step-by-step explanation:
There are 60 seconds in a minute, so the trolley travels 60 times as far as it does in one second.
(60 seconds/minute) × (10 feet/second) = 600 feet/minute
The trolley goes 600 feet in a minute.
What is the midpoint of the segment that joins the points (-12, 12) and (-6, -1)?
Answer:
[tex]\boxed {\boxed {\sf ( -9, \frac{11}{2}) }}[/tex]
Step-by-step explanation:
We are asked to find the midpoint of a segment. We essentially calculate the average of the x-coordinates and the y-coordinates using the following formula.
[tex]( \frac {x_1+x_2}{2}, \frac{ y_1 + y_2}{2})[/tex]
In this formula, (x₁ , y₁) and (x₂ , y₂) are the endpoints of the segment. For this problem, the 2 endpoints are (-12, 12) and (-6, -1). If we match the variable and the corresponding value, we see that:
x₁= -12 y₁= 12 x₂ = -6 y₂ = -1Substitute the values into the formula.
[tex]( \frac{-12 + -6}{2}, \frac{12 + -1} {2} )[/tex]
Solve the numerators.
-12 + -6 = -12 -6 = -18 12 + -1 = 12-1 = 11[tex]( \frac{-18}{2}, \frac{11}{2})[/tex]
Divide.
[tex]( -9, \frac{11}{2} )[/tex]
The midpoint of the segment is [tex]\bold {( -9, \frac{11}{2} )}[/tex].
I need help ASAP whit this question
Answer:
B. 7
Step-by-step explanation:
The original box must be 1 by 1 by 1.
So then the new box would be 2 by 2 by 2.
2×2×2=8
8-1=7
I hope this helps!
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Drag each tile to the correct box.
Using the order of operations, what are the steps for solving this expression?
40=8+32 +(15–7)x2
Arrange the steps in the order in which they are performed.
3 “squared”
15-7
14 +16
40:8
8x2
5+9
Answer:
15 - 7 (Parentheses)3² (Exponents)8 × 2 (Multiplication)40 ÷ 8 (Division)5 + 9 (Addition starting from the leftmost)14 + 16 (Addition after the first addition operation)Concept:
When encountering questions that ask for simplifying expressions through operation, following the PEMDAS method would be easier:
ParenthesesExponentsMultiplicationDivisionAdditionSubtractionTherefore, the whole process of simplifying the given expression should follow the PEMDAS method. For extra, whenever there are two occurences of the same operation, then prioritize the leftmost and go right.
Hope this helps!! :)
Please let me know if you have any questions or need further explanation
I need to know the distance between points x and y
Answer:
Step-by-step explanation:
Distance between two points P(x1,y1) and Q(x2,y2) is given by: d(P, Q) = √ (x2 − x1)2 + (y2 − y1)2 {Distance formula} 2. Distance of a point P(x, y) from the origin is given by d(0,P) = √ x2 + y2.
HELP!
10. Bill and Kate dined at a restaurant, and their bill was $49.98. They would like to leave a tip of
approximately 15% of their bill. Which of the following is the closest to this amount?
D. $12.50
E. $15.00
C. $10.00
A. $5.00
B. $7.50
Answer:
7.50$
Step-by-step explanation:
49.98÷100=0.4998=1%
0.4998×15= 7.497 which is 15%of their bill
-4x + 3x = 2
What is x
Sam opened a restaurant. On the first day he had 100 customers. On the fourth day he had 160 customers. If the number of customers per day grew linearly, what was the number of customers on the second day?
Answer:
120 customers on the 2nd day
Step-by-step explanation:
100/1ST DAY
120/2nd day
140/3rd day
160/ 4TH DAY
which choice is equivalent to the expression below? √40 + 2√10 + √90
Answer:
2√10 + 2√10 + 3√10 =7√10
Step-by-step explanation:
√40=√10×4 =√10×√4 = 2√10
√90= √10×9 = √10×√9 =3√10
Combine like terms and write the simplified
Answer:
4x-3 16t-4
Step-by-step explanation:
Answer:
1. 8x-4x=4x
5-2=3
so the answer will be
4x+3
2.9t-7t+12-8=2t+4.
3.i have no idea
4. 18x^2+12x^2+16x-11x-17+19=30x^2+5x-36.
what are the solution set of Z^2 +(2i-3)Z+5i=0
Answer:
z= 2 - 3i
1 + i
Step-by-step explanation:
I'm not sure but hope it helps.
Add or Subtract: show all written work 30 POINTS!!
Answer:
I hope it helped U
stay safe stay happy
phương trình vi phân tuyến tính cấp 1: xy' - 2y = 2x^4
Answer:
.........................
Step-by-step explanation:
............................................................................
Find the measure on z.
Answer:
z° = 135°
Step-by-step explanation:
The measure of the interior angle is 135°. The sum of the measures of its interior angles is 1080°.
I hope I helped you^_^