Integrate both sides again v = C1 4 x4 C2 Now the general solution is given as y2 = v(x)y1 y2 = ( C1 4 x4 C2)( 1 x) y2 = (C1( x3 4) C2( 1 x)) Now the first solution is given when C1 = 0, then the second solution can be when C2 = 0 which is x3 4 otherwise there are infinite solutionsby just filling the constants To checkMathx=s3t/math math\frac{d}{ds}x^2=\frac{d}{ds}(s3t)^2/math math=2(s3t)/math math\frac{d^2}{ds^2}x^2=\frac{d}{ds}(\frac{d}{ds}x^2)/math mathFunction equal to each other so there is no extra X or Y being consumed that gives no extra utility 2X=3Y rearrange Y=2X/3 – so ray from original which goes through all the corners of the L has to have the slope 2/3 The indifference curve is for when utility is 6 y 3 X 2 Ray from the origin slope is 2/3 U
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If u(x y)=x^2 y^2 2x-3xy then
If u(x y)=x^2 y^2 2x-3xy then-2x 3 y 2 4x 2 y 2 = 2x 2 y 2 • (x 2) Adding fractions that have a common denominator Adding up the two equivalent fractions Add the two equivalent fractions which now have a common denominator Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible What is the value of the expression 2x^23xy4y^2 when x = 2 and y = 4?



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I got mu(y) = e^y The point of an integrating factor is to turn an inexact differential into an exact one One physical application of this is to turn a path function into a state function in chemistry (such as dividing by T to turn q_"rev", a path function, into S, a state function, entropy) I assume that the second terms include 3y^2, not 3y (it would be odd to not simply write 9y)52 f(x;y) = C This is because the di erential equation can be written as df= 0 Here we will not develop the complete theory of exact equations, but will simply give examples of how they are dealt with Example Find the general solution to (3x2y2 3y2)dx (2x3y 6xy 3y2)dy= 0 Step 1 Check to see if M y = N x M= 3x2y2 3y2 Consider the equation x^2 2xy 4y^2 = 64 Write an expression of the slope of the curve at any point (y^p)= y prime My work 2x 2(xy^p y) 8yy^p = 0 2x 2xy^p 2y 8yy^p = 0 2xy^p 8yy^p = 2y 2x factored out y^p and You can view more similar questions or
The differential equation dy/dx xy/(1x^2) = xy^1/3 is a Bernulli equation which can be reduced to normal form taking y = V(x)^3/2 So y' = (3/2)(V^1/2)V' and the equation becomes V' (2/3)xV/(x^2–1) = 2x/3 The integrating factor is e^IntegI wonder there is a precalculus method, without using the Lagrange multiplier The function is not harmonic, so it cannot have a conjugate On the other hand, if we fix the example as u(x, y) = x3 2xy − 3xy2, then ux = 3x2 2y − 3y2 uxx = 6x uy = 2x − 6xy uyy = − 6x and this u is harmonic The harmonic conjugate v must satisfy vx = − uy vy = ux Thus vx = − 2x 6xy vy = 3x2 2y − 3y2 Hence v = − x2
Find dy/dx x2xyy^2=2 Differentiate both sides of the equation Differentiate the left side of the equation Tap for more steps Differentiate Tap for more steps By the Sum Rule, the derivative of with respect to is Differentiate using the Power Rule which states that is whereStep by step solution Consider 2x^ {2}3xy2y^ {2}2x11y12 as a polynomial over variable x Find one factor of the form kx^ {m}n, where kx^ {m} divides the monomial with the highest power 2x^ {2} and n divides the constant factor 2y^ {2}11y12 One such factor is 2xy4 Factor the polynomial by dividing it by this factorDy/dx= (2x3y1)/(3x2y1) Let x= Xp and y = Yq then dy/dx = dy/dX dy/dX= (2X2p3Y3q1)/(3X3p2Y2q1) = dy/dX = {2X3Y(2p3q1)}/{3X2Y(3p2q1)} Now 2p




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Question 3 Kanghyock Has Preferences Given By U(x,y) = 3xy 2x Y, And An Income Of 1 = 102 Prices Are Given By P = 1, Py = 2 Solve For His Optimal Consumption Bundle If x=32√2 and xy=1, then (x²3xyy²)/(x²3xyy²) 122Equations Tiger Algebra gives you not only the answers, but also the complete step by step method for solving your equations 4x^2y^2/2x^23xy2y^2^22xy/xy2x^2 so that you understand better



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Subtract 2x from each side to get y=2x2 Now substitute this expression for y into the second equation 3x2y=5 3x2(2x2)=5 solve for x 3x4x4=5x=1 ==> x=1 If x=1 then y=2(1)2=4 So(x^2–2xyy^2)dx = (y^22xyx^2)dy =>dy/dx = (x^2–2xyy^2)/((x^22xyy^2)(1) Let y = vx => dy/dx = v xdv/dx (1) becomes v xdv/dx = (x^2–2x^2vv^2x^2Solve for y x^23xyy^2=1 Move to the left side of the equation by subtracting it from both sides Use the quadratic formula to find the solutions Substitute the values , , and into the quadratic formula and solve for Simplify Tap for more steps Simplify the numerator Tap for more steps




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Integrating Factors Some equations that are not exact may be multiplied by some factor, a function u (x, y), to make them exact When this function u (x, y) exists it is called an integrating factor It will make valid the following expression ∂ (u·N (x, y)) ∂x = ∂ (u·M (x, y)) ∂yFor any unit vector, u =〈u x,u y〉let If this limit exists, this is called the directional derivative of f at the point (a,b) in the direction of u Theorem Let f be differentiable at the point (a,b) Then f has a directional derivative at (a,b) in the direction of u u = u xi u yj and D u f(a,b) = uLemma 54 Let z= x iyand suppose that f(z) = u(x;y) iv(x;y) is analytic Then the dot product of their gradients is 0, ie rurv= 0 Proof The proof is an easy application of the CauchyRiemann equations rurv= (u x;u y) (v x;v y) = u xv x u yv y= v yv x v xv y= 0 In the last step we used the CauchyRiemann equations to substitute v yfor




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Definition 161 Suppose H R2 → R has continuous second partial deriva tives on a domain D We say H is harmonic in D if for all (x,y) ∈ D, H xx(x,y)H yy(x,y) = 0 Harmonic functions arise frequently in applications, such as in the study Use the power rule, dy/dx = nx^ (x1) d y d x = n x x − 1 , on the first term 2x (3d (xy))/dx (d (y^2))/dx= (d (0))/dx 2 x 3 d ( x y) d x d ( y 2) d x = d ( 0) d x Use the product rule, (d (xy))/dx= dx/dxyxdy/dx = y xdy/dx d ( x y) d x = d x d x y x d y d x = y x d y d xMATH 106 HOMEWORK 3 SOLUTIONS 1 Using the CauchyRiemann equations, show that if f and f are both holomorphic then f is a constant Solution Let f = uiv,so f = u iv Since they are holomorphic, we can use the CauchyRiemann




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If 2x 2 3xy Y 2 X 2y 8 0 Then Dy Dx
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