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In the first part of this question you will give three different proofs of the following equation...
1. In the first part of this question you will give three different proofs of the following equat...
Please
be detailed and clear. Thanks!
1. In the first part of this question you will give three different proofs of the following equation: (arctanh (x)) = 2 da (a) () Use implicit differentiation to prove that equation () holds. ii) Use the definition of arctanh (x) via the logarithm (see p. 106 of the Lecture notes) to prove that equation () holds (ii) Use integration to prove that equation () holds (b) Using equation () from Part (a), find...
Q1. DELTA FUNCTIONS! a) Calculate the following integrals, assuming c=2 in all parts! In part iv,...
Q1. DELTA FUNCTIONS! a) Calculate the following integrals, assuming c=2 in all parts! In part iv, assume the volume V is a sphere of radius 1 centered on the origin, and the constant vector ro = (0,3,4) c 8(x-c)dx (Hint: watch out for limits of integration. Remember, c=2 throughout) 8(x-c)dx iii) Sºx-cl 8(2x)dx (Hint: see Griffiths Example 1.15) iv) SS ir-r, 8(r)dr v) M F-F, P 8°(27)di (Hint: 8' () = 8(x)O(y)8(z)) b) Evaluate the integral ſ (1+e") ( )dt...
Consider the following differential equation Note: For each part below you must give your answers in terms of fract...
Consider the following differential equation Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimals (a) The above differential equation has a snaar point at x 0 . It the singular point at x-0 is a regular singular point, then a power series for the solution y(x) can be lound using the Frobenius method. Show that x = 0 is a regular sigar point by calculating: xp(x) = y(x) = Since both...
You will integrate the following rational function: -222–162+36 dz (2-3) (x+2)(x-1) -222-162+36 To do this, start...
You will integrate the following rational function: -222–162+36 dz (2-3) (x+2)(x-1) -222-162+36 To do this, start by writing (2-3)(2+2)(x-1) A + В. 2+2 + C 2-1 1-3 (a) Put the right-hand side over a common denominator. Enter the numerator of the result. (b) Expand the numerator. It is a quadratic in z, which must equal the numerator of the original function. By equating the coefficients you should be able to formulate linear equation in A, B and C. In each...
Consider the following differential equation Note: For each part below you must give your answers in terms of fractions...
Consider the following differential equation Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimas. (a) The above differential equation has a singular point at z-0.I the singular point at z -0 is a regular singular point, then a power series for the solution ()can be found using the Frobenius method. Show that z-O is a regular singular point by calculating plz)-3 Since both of these functions are analytic at r -0...
HICULTUULUULULUI 2 Task 2 - Trapezium Rule (Maximum Mark 10) Find the equation of the line...
HICULTUULUULULUI 2 Task 2 - Trapezium Rule (Maximum Mark 10) Find the equation of the line through the points 1 - ...) a | and B(x) and find an expression for the area under this line between the points A and B. Explain carefully how your result can be used to prove the general formula for the Trapezium Rule. Notes: . This part of the assignment is testing that you can find the equation of a line and use this...
Consider the following differential equation Note: For each part below you must give your answers in terms of fractions...
Consider the following differential equation Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimals (a) The above difterential equation has a singular point at-0. If the singular point at -0 is a regular singular point, then a power series for the solution y) can be found using the Frobenius method. Show that z-0 is a regular singular point by caliculating p/a)- 2(2) Since both of these functions are analytic at -0...
Consider the following difterential equation Note: For each part below you must give your answers in terms of fractions...
Consider the following difterential equation Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimals (a) The above differential equation has a singular point at z-0.I the singular point at z-0 is a regular singular point, then a power series for the solution y)can be found using the Frobenius method. Show that z-0is a regular singular point by calculating: zr(z) = 2g() Since both of these functions are analytic at z-0 the...
Task 2 - Trapezium Rule (Maximum Mark 10) Find the equation of the line through the...
Task 2 - Trapezium Rule (Maximum Mark 10) Find the equation of the line through the points A(Y) and (.ya) and find an expression for the area under this line between the points A and B. Explain carefully how your result can be used to prove the general formula for the Trapezium Rule. Notes: • This part of the assignment is testing that you can find the equation of a line and use this to derive other formulae. Marking Criteria...
please explain the steps as well! it’s imp for me to understand this question. i have attached the table for last part...
please explain the steps as well! it’s imp for me to understand
this question. i have attached the table for last part of the
question
Consider the second order non-homogeneous constant coefficient linear ordinary differ- ential equation for y(x) ору , dy where Q(x) is a given function of r For each of the following choices of Q(x) write down the simplest choice for the particular solution yp(x) of the ODE. Your guess for yp(x) will involve some free parameters...
Please
be detailed and clear. Thanks!
1. In the first part of this question you will give three different proofs of the following equation: (arctanh (x)) = 2 da (a) () Use implicit differentiation to prove that equation () holds. ii) Use the definition of arctanh (x) via the logarithm (see p. 106 of the Lecture notes) to prove that equation () holds (ii) Use integration to prove that equation () holds (b) Using equation () from Part (a), find...
Q1. DELTA FUNCTIONS! a) Calculate the following integrals, assuming c=2 in all parts! In part iv, assume the volume V is a sphere of radius 1 centered on the origin, and the constant vector ro = (0,3,4) c 8(x-c)dx (Hint: watch out for limits of integration. Remember, c=2 throughout) 8(x-c)dx iii) Sºx-cl 8(2x)dx (Hint: see Griffiths Example 1.15) iv) SS ir-r, 8(r)dr v) M F-F, P 8°(27)di (Hint: 8' () = 8(x)O(y)8(z)) b) Evaluate the integral ſ (1+e") ( )dt...
Consider the following differential equation Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimals (a) The above differential equation has a snaar point at x 0 . It the singular point at x-0 is a regular singular point, then a power series for the solution y(x) can be lound using the Frobenius method. Show that x = 0 is a regular sigar point by calculating: xp(x) = y(x) = Since both...
You will integrate the following rational function: -222–162+36 dz (2-3) (x+2)(x-1) -222-162+36 To do this, start by writing (2-3)(2+2)(x-1) A + В. 2+2 + C 2-1 1-3 (a) Put the right-hand side over a common denominator. Enter the numerator of the result. (b) Expand the numerator. It is a quadratic in z, which must equal the numerator of the original function. By equating the coefficients you should be able to formulate linear equation in A, B and C. In each...
Consider the following differential equation Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimas. (a) The above differential equation has a singular point at z-0.I the singular point at z -0 is a regular singular point, then a power series for the solution ()can be found using the Frobenius method. Show that z-O is a regular singular point by calculating plz)-3 Since both of these functions are analytic at r -0...
HICULTUULUULULUI 2 Task 2 - Trapezium Rule (Maximum Mark 10) Find the equation of the line through the points 1 - ...) a | and B(x) and find an expression for the area under this line between the points A and B. Explain carefully how your result can be used to prove the general formula for the Trapezium Rule. Notes: . This part of the assignment is testing that you can find the equation of a line and use this...
Consider the following differential equation Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimals (a) The above difterential equation has a singular point at-0. If the singular point at -0 is a regular singular point, then a power series for the solution y) can be found using the Frobenius method. Show that z-0 is a regular singular point by caliculating p/a)- 2(2) Since both of these functions are analytic at -0...
Consider the following difterential equation Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimals (a) The above differential equation has a singular point at z-0.I the singular point at z-0 is a regular singular point, then a power series for the solution y)can be found using the Frobenius method. Show that z-0is a regular singular point by calculating: zr(z) = 2g() Since both of these functions are analytic at z-0 the...
Task 2 - Trapezium Rule (Maximum Mark 10) Find the equation of the line through the points A(Y) and (.ya) and find an expression for the area under this line between the points A and B. Explain carefully how your result can be used to prove the general formula for the Trapezium Rule. Notes: • This part of the assignment is testing that you can find the equation of a line and use this to derive other formulae. Marking Criteria...
please explain the steps as well! it’s imp for me to understand
this question. i have attached the table for last part of the
question
Consider the second order non-homogeneous constant coefficient linear ordinary differ- ential equation for y(x) ору , dy where Q(x) is a given function of r For each of the following choices of Q(x) write down the simplest choice for the particular solution yp(x) of the ODE. Your guess for yp(x) will involve some free parameters...
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