The Riccati equation is one of the most interesting nonlinear differential equations of first order It's written in the form where a (x), b (x), c (x) are continuous functions of x The Riccati equation is used in different areas of mathematics (for example, in algebraic geometry and the theory of conformal mapping), and physics It alsoStandard equation of a parabola that opens left and symmetric about xaxis with vertex at origin y 2 = 4ax Standard equation of a parabola that opens up and symmetric about xaxis with at vertex (h, k) (y k) 2 = 4a(x h) Graph of y 2 = 4axThe coefficient of xy2 in equals because there are three x,y strings of length 3 with exactly two y s, namely, corresponding to the three 2element subsets of {1, 2, 3}, namely, where each subset specifies the positions of the y in a corresponding string General case
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Plot x2+(y-^x)2=1 formula- sin (π/2 – x) = cos x cos (π/2 – x) = sin x sin (π/2 x) = cos x cos (π/2 x) = – sin x sin (3π/2 – x) = – cos x cos (3π/2 – x) = – sin x sin (3π/2 x) = – cos x cos (3π/2 x) = sin x sin (π – x) = sin x cos (π – x) = – cos x sin (π x) = – sin xExtended Keyboard Examples Upload Random Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music
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Algebra Expansion Formulae, Mathematics Mathematics Menu The following are algebraix expansion formulae of selected polynomials Square of summationX y is a binomial in which x and y are two terms In mathematics, the cube of sum of two terms is expressed as the cube of binomial x y It is read as x plus y whole cube It is mainly used in mathematics as a formula for expanding cube of sum of any two terms in their terms ( x y) 3 = x 3 y 3 3 x 2 y 3 x y 2Distance Formula Given the two points (x 1, y 1) and (x 2, y 2), the distance d between these points is given by the formula Don't let the subscripts scare you They only indicate that there is a "first" point and a "second" point;
0 Mithra, added an answer, on 23/9/ Mithra answered this (xyz) 2 = x 2 y 2 z 2 2xy 2yz2zx Was this answer helpful?A Differentiation formulas list has been provided here for students so that they can refer to these to solve problems based on differential equations This is one of the most important topics in higher class Mathematics The general representation of the derivative is d/dx This formula list includes derivatives for constant, trigonometric functions, polynomials, hyperbolic, logarithmic For tan (x y), numerator is positive & denominator is negative For tan (x – y), numerator is negative & denominator is positive Let's take x = 60°, y = 30° and verify sin (x y) = sin x cos y cos x sin y sin (60° 30°) = sin 60° cos 30° cos 60° sin 30° sin (90°) = (√3/2) × (√3/2) (1/2) × (1/2) 1 = 3/4 1/4 1 = 4/4 1 = 1 Hence verified sin (x – y) = sin x cos y – cos x sin
Linear equation y= x distance PQ slope θ When the equation becomes parallel to yaxis, it is displayed as infinity (∞) L i n e a r e q u a t i o n t h r o u g h P a n d Q P (x 1, y 1), Q (x 2, y 2) x 2 x y y 2 = ( x y / 2) 2 3 y 2 4 and you can't really do better since the lefthand side doesn't factor Even if the expression doesn't factor, completing the square is useful in solving quadraticlike equations, for instance 9 sin 2 x 12 sin x − 8 = 0 You can partially factor this as ( 3 sin x 2) 2 − 12 = 0Hence, the first cos 2X formula follows, as \(\cos 2X = \cos ^{2}X – \sin ^{2}X\) And for this reason, we know this formula as double the angle formula, because we are doubling the angle Other Formulae of cos 2X \(\cos 2X = 1 – 2 \sin ^{2}X \) To derive this, we need to start from the earlier derivation As we already know that,
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In any triangle we have 1 The sine law sin A / a = sin B / b = sin C / c 2 The cosine laws a 2 = b 2 c 2 2 b c cos A b 2 = a 2 c 2 2 a c cos B c 2 = a 2 b 2 2 a b cos C Relations Between Trigonometric Functions 461K people helped that is the formula of x 2y2= (xy) (xy) acobdarfq and 363 more users found this answer helpful heart outlined Thanks 0 star star star starFree PreAlgebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators stepbystep
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Example Plot (x−4) 2 (y−2) 2 = 25 The formula for a circle is (x−a) 2 (y−b) 2 = r 2 So the center is at (4,2) And r 2 is 25, so the radius is √25 = 5 So we can plot The Center (4,2) Up (4,25) = (4,7) Down (4,2−5) = (4,−3) Left (4−5,2) = (−1,2) Right (45,2) = (9,2) When I set y ( x) = A x u ( x) = A u x , (I think this is how you solve second order with variable coefficients) I got to the point u ″ ( x − x 3) 2 u ′ ( 1 − 2 x 2) = u ″ ( x − x 3) 2 u ′ ( 1 − 3 x 2) 2 u ′ x 2 = d d x u ′ ( x − x 3) 2 u ′ ( 1 − x 2) = 0 Example 7 (y 1) 2 = x If we think about the equation (y 1) 2 = x for a while, we can see that x will be positive for all values of y (since any value squared will be positive) except y = −1 (at which point x = 0) In the equation (y 1) 2 = x, the "plus 1" in brackets has the effect of moving our rotated parabola down one unit Example
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To find out the slope of a line, we need only two points from that line, (x 1, y 1) and (x 2, y 2) There are three steps for calculating the slope of a straight line Step1 Identify two points on the line Step2 Select one to be (x 1,y 1) and the other to be(x 2,y 2) Step3 Use the slope of the line formula to calculate the slopeThat is, that you have two points Whichever one you call "first" or "second" is up to youAlgebra Factor x^2y^2 x2 − y2 x 2 y 2 Since both terms are perfect squares, factor using the difference of squares formula, a2 −b2 = (ab)(a−b) a 2 b 2 = ( a b) ( a b) where a = x a = x and b = y b = y (xy)(x−y) ( x y) ( x y)
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Y x = y 2 y 1 x 2 x 1 = f(x 2) f(x 1) x 2 x 1 (61) It's a linear approximation of the behavior of f between the points x 1 and x 2 7 Quadratic Functions The quadratic function (aka the parabola function or the square function) f(x) = ax2 bx c (71) can always be written in the form f(x) =As you indicated the parabola x = y 2 is "on its side" x = y 2 You can determine the shape of x = 4 y 2 by substituting some numbers as you suggest Sometimes you can see what happens without using specific points Suppose the curves are x = y 2 and x = 4 y 2 and and you want to find points on the two curves with the same yvalue Then substitute y 2 from the first equation into theMath Input Use Math Input Mode to directly enter textbook math notation Try it
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Math mode is susceptible to linebreaks, but you can use braces to prevent that the formula${(xy)^2}$ will not break across lines, while $(xy)^2$ may Share Improve this answer Follow answered at 131 Mike Stay MikeWell, my first goto was to do this \displaystyle x^2 xy y^2 = 0 \displaystyle (x^2 xy y^2)(x y) = 0 (any number multiplied by zero is equal to zero) \displaystyle x^3 y^3 = 0 However, there's one issue xBy solving these two equations we can find the intersection of two lines formula The formula for the point of intersection of two lines will be as follows x = b 1 c 2 − b 2 c 1 a 1 b 2 − a 2 b 1 y = c 1 a 2 − c 2 a 1 a 1 b 2 − a 2 b 1
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For more calculus solutions, algebra help, please see wwwblackpenredpencomintegral of 1/(x^2a^2)First type the equation 2x3=15 Then type the @ symbol Then type x=6 Try it now 2x3=15 @ x=6 Clickable Demo Try entering 2x3=15 @ x=6 into the text box After you enter the expression, Algebra Calculator will plug x=6 in for the equation 2x3=15 2(6)3 = 15 The calculator prints "True" to let you know that the answer is right More ExamplesThe equation is in standard form 2x=y2 2 x = y 2 Divide both sides by 2 Divide both sides by 2 \frac {2x} {2}=\frac {y2} {2} 2 2 x = 2 y 2 Dividing by 2 undoes the multiplication by 2 Dividing by 2 undoes the multiplication by 2
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Rewrite as x^22xy=0 This is a quadratic equation in variable x Don't be confused, I'm just pointing out that we will temporarily be thinking of y as a constant (a number) We would solve by factoring if we could, but we can't so we'll use the quadratic formula, which says that the solutions to 2x^2 bx c = 0 are x=(bsqrt(b^24ac))/(2a)Algebra Calculator MathPapa Algebra Calculator What do you want to calculate?X 2 y 2 = r 2 This is true for any point on the circle since any point on the circle is an equal distance, r, from the center Centered at any location To find the equation for a circle in the coordinate plane that is not centered at the origin, we use the distance formula
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Examples 12 , 1/31/4 , 2^3 * 2^2 (x1) (x2) (Simplify Example), 2x^22y @ x=5, y=3 (Evaluate Example) y=x^21 (Graph Example), 4x2=2 (x6) (Solve Example) Algebra Calculator is a calculator that gives stepbystep help on algebra problemsGraph of xy = yx The line and curve intersect at ( e, e ) In general, exponentiation fails to be commutative However, the equation x y = y x {\displaystyle x^ {y}=y^ {x}} holds in special cases, such as x = 2 , y = 4 {\displaystyle x=2,\ y=4}With origin O (0,0) as centre draw a circle of radius r Take any point P (x,y) on the circle Then OP = r OP^2 = r^2 Applying distance formula we get (x0)^2 (y0)^2 = r^2 ie, x^2 y^2 = r^2 Hence the equation of the circle with centre at the origin and radius r is x^2 y^2 = r^2 23K views View upvotes Related Answer Dave Benson
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When we have a sum (difference) of two or three numbers to power of 2 or 3 and we need to remove the brackets we use polynomial identities (short multiplication formulas) (x y) 2 = x 2 2xy y 2 (x y) 2 = x 2 2xy y 2 Example 1 If x = 10, y = 5a (10 5a) 2 = 10 2 2·10·5a (5a) 2 = 100 100a 25a 2Y x = y 2 y 1 x 2 x 1 = rise run Linear Function/Slopeintercept form This graph is a line with slope m and y intercept(0;b) y= mx b or f(x) = mx b PointSlope form The equation of the line passing through the point (x 1;y 1) with slope mis y= m( x 1) y 1 Quadratic Functions and Formulas Examples of Quadratic Functions x y y= x2Answer (1 of 16) YES, there are solutions, an infinite amount in fact!
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The hint x^2 4y^2 = (x 2y)(x 2y) means that n must be divisible by (x 2y) and (x 2y)So, for example an integer factorization of n yields a relatively small set of integers that can be searched for whether they can produce numbers of the form (x 2y) and (x 2y) It suffices to consider positive integers x and y because due the squares in the equation, for every x also x isX and Y coordinates is an address, which helps to locate a point in twodimensional space Any point in the coordinate plane is referred by a point (x, y), where the x value is the position of the point with reference to the xaxis, and the y value is the position ofAn outline of Isaac Newton's original discovery of the generalized binomial theorem Many thanks to Rob Thomasson, Skip Franklin, and Jay Gittings for their
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(xy)^2=(xy)(xy)=x{\color{#D61F06}{yx}} y=x{\color{#D61F06}{xy}}y=x^2 \times y^2\ _\square (x y) 2 = (x y) (x y) = x y x y = x x y y = x 2 × y 2 For noncommutative operators under some algebraic structure, it is not always true Let Q \mathbb Q Q be the set of quaternions, and let x = i, y = j ∈ Q x=i,y=j\in\mathbb Q x = i, y = j ∈ QFree math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with stepbystep explanations, just like a math tutorAll equations of the form a x 2 b x c = 0 can be solved using the quadratic formula 2 a − b ± b 2 − 4 a c The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction 2x^ {2}\left (2y\right)x=0 2 x 2 ( − 2 y) x =
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It can be factored as x^2 y^2 = (xy)(xy) Notice that when you multiply (xy) by (xy) then the terms in xy cancel out, leaving x^2y^2 (xy)(xy) = x^2xyyxy^2 = x^2xyxyy^2 = x^2y^2 In general, if you spot something in the form a^2b^2 then it can be factored as (ab)(ab) For example 9x^216y^2 = (3x)^2(4y)^2 = (3x4y)(3x4y)Begin completing the square Move the constant term to the right Add 'y 2 ' to each side of the equation xy x 2 1y 2 y 2 = 0 y 2 Combine like terms 1y 2 y 2 = 0 xy x 2 0 = 0 y 2 xy x 2 = 0 y 2 Remove the zero xy x 2 = y 2 The x term is xy Take half its coefficient (05y) Square it (025y 2) and add it to both sidesX^2y^2=9 (an equation of a circle with a radius of 3) sin (x)cos (y)=05 2x−3y=1 cos (x^2)=y (x−3) (x3)=y^2 y=x^2 If you don't include an equals sign, it will assume you mean " =0 " It has not been well tested, so have fun with it, but don't trust it If it gives you problems, let me know
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If you factor the second equation and solve for x in terms of y, you get x2 2 xy – 8 y2 = 0 ( x 4 y ) ( x – 2 y) = 0 x 4 y = 0 or x – 2 y = 0 x = –4 y or x = 2 y Plug these into the first equation for y, and solve for the x values This last example (the first way I worked it) is about as complicated as these things ever get
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