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Problem 1: Recall that the general solution for the potential with cylindrical symmetry is given by...
In spherical polar coordinates (r, 0, ¢), the general solution of Laplace's equation which has cylindrical...
In spherical polar coordinates (r, 0, ¢), the general solution of Laplace's equation which has cylindrical symmetry about the polar axis is bounded on the polar axis can be expressed as u(r, 0) = Rm(r)P,(cos 0), (A) where P is the Legendre polyomial of degree n, and R(r) is the general solution of the differential equation *() - n(n + 1)R = 0, (r > 0), dr dr where n is a non-negative integer. (You are not asked to show...
Finding Unknown Constants Given Boundary Conditions My Solutions Recall that when we find the ant...
Need help what is the answers
Finding Unknown Constants Given Boundary Conditions My Solutions Recall that when we find the antiderivative of a function, we include a C at the end of the answer since this will not change the derivative of the answer (the derivative of any constant is zero). Often, we are given a second derivative with specific boundary conditions which allow us to find the unknown constants by substituting the values for x and y into the...
the below is the previous question solution: 1. Recall the following boundary-value problem on the interval [0, 1] from...
the below is the previous question solution:
1. Recall the following boundary-value problem on the interval [0, 1] from Homework 2: f" =-Xf, f'(1) =-f(1). f(0) = 0, Show that if (Anh) and to this boundary-value problem, λι, λ2 〉 0, λιメÂn then fi and f2 are orthogonal with respect to the standard inner product (.9)J( gr)dr. (You may use the solution posted on the course website, or work directly from the equation and boundary conditions above.) (λ2'J2) are two...
Physics 102 Extra Credit Legendre Polynomials Problem The following problem is worth 5 ertra credit points!...
Physics 102 Extra Credit Legendre Polynomials Problem The following problem is worth 5 ertra credit points! Consider a disk of radius R carrying charge q (un formly distributed) and lying in the ry plane as seen in the diagram. We want to determine the potential V(r,0) everywhere outside the disk, for r R (because of the azimuthal symmetry the potential doesnt depend on φ). We have seen earlier that the potential along the z-axis (when 0-0) is gr R2 V(ro-ro...
1 Potential of concentric spheres A spherical shell with internal radius Rį and external radius R2...
1 Potential of concentric spheres A spherical shell with internal radius Rį and external radius R2 has a potential in its surfaces given by 0(R1,0,0) = Vi sin (20) sin(0) and (R2,0,0) = V2 sin (20) sin(0) (V1 and V2 are constants). If there are not electric charges any where inside or outside the shell R R2 (a) Write the general solution for the electric potential o in each of the three regions of interest: r < R1, R; <r...
5. Consider a long rectangular "gutter" of length a in the x direction and infi- nite...
5. Consider a long rectangular "gutter" of length a in the x direction and infi- nite height in the y direction. The gutter is infinitely long in the z direction, so the potential V inside the gutter only depends on x andy. The left (x-0,y), and right (r- a,y) sides of the gutter are grounded so that the potential V(x,y) is zero on those surfaces. The bottom surface of the gutter is kept fixed at a potential given by V(r,y-0)-...
solution for all 4 please In Problems 1-3, solve the given DE or IVP (Initial-Value Problem)....
solution for all 4 please
In Problems 1-3, solve the given DE or IVP (Initial-Value Problem). [First, you need to determine what type of DE it is. 1. (2xy + cos y) dx + (x2 – x sin y – 2y) dy = 0. 1 dy 2. + cos2 - 2.cy y(y + sin x), y(0) = 1. + y2 dc 3. [2xy cos (2²y) – sin x) dx + x2 cos (x²y) dy = 0. (1+y! x" y® is...
Problem 1. Find the general solution of an 1D heat equation: T(x, t) = 4Txx(x, t)...
Problem 1. Find the general solution of an 1D heat equation: T(x, t) = 4Txx(x, t) with the boundary conditions T(0,t) = T(2,t) = 0. Note that T(x,t) denotes the temperature profile along x of a uniform rod of length 2. Problem 2. Solve the following 1D wave equation: 0ct(x, t) = 0xx(x, t) with the boundary conditions 0(0,t) = 0,(1,t) = 0, where 8(x, t) refers to the twist angle of a uniform rod of unit length. Problem 3....
Consider a medical device where blood is circulated in the annular space between two coaxial cyli...
do
the second prob pic
Consider a medical device where blood is circulated in the annular space between two coaxial cylinders (Figure 1). The inner cylinder (radius cylinder (radius R) is rotating with constant anacibeNewtonian fluid (density o. are infinitely long, and that blood behaves as an tncompcessiole viscosity . Ignore the effect of gravity. whereas the outer velocity oAssume that the cylinders 1a. Write a conservation equations appropriate to determine the fluid velocity profile insido the annular gap, along...
In Problems 1-3, solve the given DE or IVP (Initial-Value Problem). [First, you need to determine...
In Problems 1-3, solve the given DE or IVP (Initial-Value Problem). [First, you need to determine what type of DE it is.) 1. (2xy + cos y) dx + (x2 – 2 siny – 2y) dy = 0. 2. + cos2 - 2ary dy dar y(y +sin x), y(0) = 1. 1+ y2 3. [2ry cos (x²y) - sin r) dx + r?cos (r?y) dy = 0. 4. Determine the values of the constants r and s such that (x,y)...
In spherical polar coordinates (r, 0, ¢), the general solution of Laplace's equation which has cylindrical symmetry about the polar axis is bounded on the polar axis can be expressed as u(r, 0) = Rm(r)P,(cos 0), (A) where P is the Legendre polyomial of degree n, and R(r) is the general solution of the differential equation *() - n(n + 1)R = 0, (r > 0), dr dr where n is a non-negative integer. (You are not asked to show...
Need help what is the answers
Finding Unknown Constants Given Boundary Conditions My Solutions Recall that when we find the antiderivative of a function, we include a C at the end of the answer since this will not change the derivative of the answer (the derivative of any constant is zero). Often, we are given a second derivative with specific boundary conditions which allow us to find the unknown constants by substituting the values for x and y into the...
the below is the previous question solution:
1. Recall the following boundary-value problem on the interval [0, 1] from Homework 2: f" =-Xf, f'(1) =-f(1). f(0) = 0, Show that if (Anh) and to this boundary-value problem, λι, λ2 〉 0, λιメÂn then fi and f2 are orthogonal with respect to the standard inner product (.9)J( gr)dr. (You may use the solution posted on the course website, or work directly from the equation and boundary conditions above.) (λ2'J2) are two...
Physics 102 Extra Credit Legendre Polynomials Problem The following problem is worth 5 ertra credit points! Consider a disk of radius R carrying charge q (un formly distributed) and lying in the ry plane as seen in the diagram. We want to determine the potential V(r,0) everywhere outside the disk, for r R (because of the azimuthal symmetry the potential doesnt depend on φ). We have seen earlier that the potential along the z-axis (when 0-0) is gr R2 V(ro-ro...
1 Potential of concentric spheres A spherical shell with internal radius Rį and external radius R2 has a potential in its surfaces given by 0(R1,0,0) = Vi sin (20) sin(0) and (R2,0,0) = V2 sin (20) sin(0) (V1 and V2 are constants). If there are not electric charges any where inside or outside the shell R R2 (a) Write the general solution for the electric potential o in each of the three regions of interest: r < R1, R; <r...
5. Consider a long rectangular "gutter" of length a in the x direction and infi- nite height in the y direction. The gutter is infinitely long in the z direction, so the potential V inside the gutter only depends on x andy. The left (x-0,y), and right (r- a,y) sides of the gutter are grounded so that the potential V(x,y) is zero on those surfaces. The bottom surface of the gutter is kept fixed at a potential given by V(r,y-0)-...
solution for all 4 please
In Problems 1-3, solve the given DE or IVP (Initial-Value Problem). [First, you need to determine what type of DE it is. 1. (2xy + cos y) dx + (x2 – x sin y – 2y) dy = 0. 1 dy 2. + cos2 - 2.cy y(y + sin x), y(0) = 1. + y2 dc 3. [2xy cos (2²y) – sin x) dx + x2 cos (x²y) dy = 0. (1+y! x" y® is...
Problem 1. Find the general solution of an 1D heat equation: T(x, t) = 4Txx(x, t) with the boundary conditions T(0,t) = T(2,t) = 0. Note that T(x,t) denotes the temperature profile along x of a uniform rod of length 2. Problem 2. Solve the following 1D wave equation: 0ct(x, t) = 0xx(x, t) with the boundary conditions 0(0,t) = 0,(1,t) = 0, where 8(x, t) refers to the twist angle of a uniform rod of unit length. Problem 3....
do
the second prob pic
Consider a medical device where blood is circulated in the annular space between two coaxial cylinders (Figure 1). The inner cylinder (radius cylinder (radius R) is rotating with constant anacibeNewtonian fluid (density o. are infinitely long, and that blood behaves as an tncompcessiole viscosity . Ignore the effect of gravity. whereas the outer velocity oAssume that the cylinders 1a. Write a conservation equations appropriate to determine the fluid velocity profile insido the annular gap, along...
In Problems 1-3, solve the given DE or IVP (Initial-Value Problem). [First, you need to determine what type of DE it is.) 1. (2xy + cos y) dx + (x2 – 2 siny – 2y) dy = 0. 2. + cos2 - 2ary dy dar y(y +sin x), y(0) = 1. 1+ y2 3. [2ry cos (x²y) - sin r) dx + r?cos (r?y) dy = 0. 4. Determine the values of the constants r and s such that (x,y)...
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