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Homogeneous Function Show that if f(x, y) is homogeneous of degree n, then
[Hint: Let
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Chapter 13 Solutions
Multivariable Calculus
- a function is defined on the closed interval[-4,8] consists of two linear pieces and a semi circle defined by f(x)=3x integral from 0 to x Ā g(t) dtĀ find f(7) and f'(7) and find the value of x in the closed interval[-4,] at which f attains its maximum value.Ā Justify your answerarrow_forwardExistence. Integrate the function f(x, y) = 1/(1 - xĀ²- yĀ²) over the disk xĀ²+ yĀ² ā¤ 3/4. Does the integral of f(x, y) exist over the disk xĀ²+ yĀ² ā¤ 1? Justify your answer.arrow_forwardHeat flux Suppose a solid object in ā3 has a temperature distribution given by T(x, y, z). The heat flow vector field in the object is F = -kāT, where the conductivity k > 0 is a property of the material. Note that the heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector isā ā F = -kāā āT = -kā2T (the Laplacian of T). Compute the heat flow vector field and its divergence for the following temperature distributions.arrow_forward
- Heat flux Suppose a solid object in ā3 has a temperature distribution given by T(x, y, z). The heat flow vector field in the object is F = -kāT, where the conductivity k > 0 is a property of the material. Note that the heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector isā ā F = -kāā āT = -kā2T (the Laplacian of T). Compute the heat flow vector field and its divergence for the following temperature distributions. T(x, y, z) = 100e-x2 + y2 + z2arrow_forward(a) Show that ā«-āāf(x) dxĀ is divergent.(b) Show that Iim ā«-t t x dx = 0 tāāĀ Ā This shows that we canāt define ā«-āāf(x) dxĀ = Iim ā«-t t f(x) dx = 0 tāāarrow_forwardDensity and mass Suppose a thin rectangular plate, represented by aregion R in the xy-plane, has a density given by the function Ļ(x, y);this function gives the area density in units such as grams per squarecentimeter (g/cm2). The mass of the plate is ā«ā«R Ļ(x, y) dA. AssumeR = {(x, y): 0 ā¤ x ā¤ Ļ/2, 0 ā¤ y ā¤ Ļ} and find the mass ofthe plates with the following density functions.a. Ļ(x, y) = 1 + sin xĀ Ā Ā Ā Ā Ā Ā Ā b. Ļ(x, y) = 1 + sin yc. Ļ(x, y) = 1 + sin x sin yarrow_forward
- Homogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degreeĀ nĀ if f(Źx, Źy) = Źnf(x,y) Ā Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v =Ā x/y to have the GS in terms of x and y have C which is arbitrary constant Ā Solve the DE. (Homogeneous DE) 1. xdx + sin2 (y/x) [ydx - xdy = 0]arrow_forwardHomogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degreeĀ nĀ if f(Źx, Źy) = Źnf(x,y) Ā Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v =Ā x/y to have the GS in terms of x and y have C which is arbitrary constant Ā Solve the DE. (Homogeneous DE) 1. x2y' = 4x2 + 7xy + 2y2arrow_forwardHomogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degreeĀ nĀ if f(Źx, Źy) = Źnf(x,y) Ā Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v =Ā x/y to have the GS in terms of x and y have C which is arbitrary constant Ā Solve the DE. (Homogeneous DE) 1. [y - (x2+y2)1/2]dx - xdy = 0arrow_forward
- Homogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degreeĀ nĀ if f(Źx, Źy) = Źnf(x,y) Ā Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v =Ā x/y to have the GS in terms of x and y have C which is arbitrary constant Ā Solve the DE. (Homogeneous DE) 1. (x3 + y3) dx + 3xy2dy = 0arrow_forwardHomogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degreeĀ n if f(Źx, Źy) = Źnf(x,y) Ā Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v =Ā x/y to have the GS in terms of x and y have C which is arbitrary constant Ā Solve the DE. (Homogeneous DE) 1. (y2-x2)dx + xydy = 0arrow_forwardHomogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degreeĀ nĀ if f(Źx, Źy) = Źnf(x,y) Ā Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v =Ā x/y to have the GS in terms of x and y have C which is arbitrary constant Ā Solve the DE. (Homogeneous DE) 1. [xcos2(y/x) - y]dx + xdy = 0arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage