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plz print ur answer Question2 Consider the differential equation We saw in class that one solution is the Bessel functi...
Question 2 Consider the differential equation We saw in class that one solution is the Bessel function (a) Suppose we h...
Question 2 Consider the differential equation We saw in class that one solution is the Bessel function (a) Suppose we have a solution to this ODE in the form y-Σχ0CnXntr where cn 0. By considering the first term of this series show that r must satisfy r2-4-0 (and hence that r = 2 or r =-2) (b) Show that any solution of the form y-ca:0G,2n-2 must satisfy C0 (c) From the theory about singular solutions we know that a linearly...
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the s...
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the series converges for any E (-8,s), s > 0 Substitute the series solution y in to the differential equation and simplify the terms to obtain an expression of the form 1-1 where f(r) is a polynomial of degree 2. (c) Determine the values of r....
(15 pts) Bessel functions and the vibration of a circular drum In polar coordinates, the Laplacia...
(15 pts) Bessel functions and the vibration of a circular drum In polar coordinates, the Laplacian is just like the Laplacian for the cylinder, but with the removed part เอ The structure of the Laplacian is what we call separable because the r and 0 terms are separate this allows us to solve certain physics problems on the disc by searching for solutions of the form f(r,0)-ar)b() The vibration of a circular drum head is described by 02t where u...
5. In class we saw that the function r(u, v) = (sin u, (2 + cos...
5. In class we saw that the function r(u, v) = (sin u, (2 + cos u) cos v, (2 + cos u) sin v), 0<u<27, 050521 parametrizes a torus T, which is depicted below. (a) Calculate ||ru x rull. (b) Show that T is smooth. (c) Find the equation of the tangent plane to T at (0,). (d) Find the surface area of T (e) Earlier in the semester, we observed that a torus can be built out of...
(1 point) In this problem you will solve the differential equation or @() (1) Since P(a) 0 are not analytic at and 2()...
(1 point) In this problem you will solve the differential equation or @() (1) Since P(a) 0 are not analytic at and 2() is a singular point of the differential equation. Using Frobenius' Theorem, we must check that are both analytic a # 0. Since #P 2 and #2e(z) are analytic a # 0-0 is a regular singular point for the differential equation 28x2y® + 22,23, + 4y 0 From the result ol Frobenius Theorem, we may assume that 2822y"...
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...
fill in the blank. Calc 2 Part a and b Consider the following differential equation by"...
fill in the blank. Calc 2 Part a and b
Consider the following differential equation by" + 3y - 3y = 0 Exercise (a) satisfy the equation? For what values of r does the function y = e Step 1 in the To determine the values of r for which er satisfies the differential equation, we substitute f(x) = e equation, 67"(x) + 3f'(x) - 3f(x) = 0. Thus, we need to find f'(x) and "(x). f'(X) = Click here...
+ (3) ar2 2. Recall from lectures that the governing PDE for vibrations of a circular...
+ (3) ar2 2. Recall from lectures that the governing PDE for vibrations of a circular drum lid is 1 au 1 ay c? + 012 72 302 for r € (0,R), 0€ (-2,7), and t > 0, and the boundary condition is (R, 6,t) = 0 for t>0 and -150<7. rar (4) You will search for a solution of the form v(r,0,t) = G(r) sin(30) cos(w t), (5) for a function G that satisfies the ODE m2 G" +rG'...
As a specific example we consider the non-homogeneous problem y"+9y sec (3) (1) The general solution...
As a specific example we consider the non-homogeneous problem y"+9y sec (3) (1) The general solution of the homogeneous problem (called the complementary solution, sab2) is gliven in terms of a pair of linearly independent solutions, y1W Here α and b are arbitrary constants. Find a fundamental set for y"+9y -0 and enter your results as a comma separated list BEWARE Ntice that the above set does not require you to decide which function is to be called y or...
Problem 1: Consider a 2nd order homogeneous differential equation of the form aa2y"(x)bay(x) + cy =...
Problem 1: Consider a 2nd order homogeneous differential equation of the form aa2y"(x)bay(x) + cy = 0 (1) where a, b, c are constants satisfy so that y(x) = x (a) Find and justify what conditions should a constant m to (1) is a solution (b) Using your solution to (1) Write these three different cases as an equation that a, b,c satisfy. Hint: Use the quadratic formula we should get three different cases for the values that m can...
Question 2 Consider the differential equation We saw in class that one solution is the Bessel function (a) Suppose we have a solution to this ODE in the form y-Σχ0CnXntr where cn 0. By considering the first term of this series show that r must satisfy r2-4-0 (and hence that r = 2 or r =-2) (b) Show that any solution of the form y-ca:0G,2n-2 must satisfy C0 (c) From the theory about singular solutions we know that a linearly...
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the series converges for any E (-8,s), s > 0 Substitute the series solution y in to the differential equation and simplify the terms to obtain an expression of the form 1-1 where f(r) is a polynomial of degree 2. (c) Determine the values of r....
(15 pts) Bessel functions and the vibration of a circular drum In polar coordinates, the Laplacian is just like the Laplacian for the cylinder, but with the removed part เอ The structure of the Laplacian is what we call separable because the r and 0 terms are separate this allows us to solve certain physics problems on the disc by searching for solutions of the form f(r,0)-ar)b() The vibration of a circular drum head is described by 02t where u...
5. In class we saw that the function r(u, v) = (sin u, (2 + cos u) cos v, (2 + cos u) sin v), 0<u<27, 050521 parametrizes a torus T, which is depicted below. (a) Calculate ||ru x rull. (b) Show that T is smooth. (c) Find the equation of the tangent plane to T at (0,). (d) Find the surface area of T (e) Earlier in the semester, we observed that a torus can be built out of...
(1 point) In this problem you will solve the differential equation or @() (1) Since P(a) 0 are not analytic at and 2() is a singular point of the differential equation. Using Frobenius' Theorem, we must check that are both analytic a # 0. Since #P 2 and #2e(z) are analytic a # 0-0 is a regular singular point for the differential equation 28x2y® + 22,23, + 4y 0 From the result ol Frobenius Theorem, we may assume that 2822y"...
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...
fill in the blank. Calc 2 Part a and b
Consider the following differential equation by" + 3y - 3y = 0 Exercise (a) satisfy the equation? For what values of r does the function y = e Step 1 in the To determine the values of r for which er satisfies the differential equation, we substitute f(x) = e equation, 67"(x) + 3f'(x) - 3f(x) = 0. Thus, we need to find f'(x) and "(x). f'(X) = Click here...
+ (3) ar2 2. Recall from lectures that the governing PDE for vibrations of a circular drum lid is 1 au 1 ay c? + 012 72 302 for r € (0,R), 0€ (-2,7), and t > 0, and the boundary condition is (R, 6,t) = 0 for t>0 and -150<7. rar (4) You will search for a solution of the form v(r,0,t) = G(r) sin(30) cos(w t), (5) for a function G that satisfies the ODE m2 G" +rG'...
As a specific example we consider the non-homogeneous problem y"+9y sec (3) (1) The general solution of the homogeneous problem (called the complementary solution, sab2) is gliven in terms of a pair of linearly independent solutions, y1W Here α and b are arbitrary constants. Find a fundamental set for y"+9y -0 and enter your results as a comma separated list BEWARE Ntice that the above set does not require you to decide which function is to be called y or...
Problem 1: Consider a 2nd order homogeneous differential equation of the form aa2y"(x)bay(x) + cy = 0 (1) where a, b, c are constants satisfy so that y(x) = x (a) Find and justify what conditions should a constant m to (1) is a solution (b) Using your solution to (1) Write these three different cases as an equation that a, b,c satisfy. Hint: Use the quadratic formula we should get three different cases for the values that m can...
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