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Question 2 Consider the differential equation We saw in class that one solution is the Bessel function (a) Suppose we h...
plz print ur answer Question2 Consider the differential equation We saw in class that one solution is the Bessel functi...
plz print ur answer
Question2 Consider the differential equation We saw in class that one solution is the Bessel function (-1)" ( 2n+2 2+n)! 2) n=0 (a) Suppose we have a solution to this ODE in the form y = Σ。:0cmFn+r where 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-must satisfy co c) From the theory about...
1. (a) Write down a second-order differential equation such that y(x) = 3xe-20 is a solution....
1. (a) Write down a second-order differential equation such that y(x) = 3xe-20 is a solution. (b) Determine whether the pair of functions y1 (2) = x sin(x) and y2(x) = cos(x) are linearly inde- pendent. Justify your answer. (c) Write down a second-order ODE which has a solution y(x) that is bounded for all 3 (i.e. y(x) < for all 2) but y(x) is not a constant function.
ODE problem Implicit equation.2 Consider the ODE (a) Solve the equation by first obtaining explicit equation(s)...
ODE problem
Implicit equation.2 Consider the ODE (a) Solve the equation by first obtaining explicit equation(s) for y. Here y : R → R is a scalar function. Your answer should be a family of solution parameterized by a single (b) Show that the function y(x) 0 also solves the equation even though it does not belong (c) Sketch a few representative functions of the family found in part (a) together with the parameter (an integration constant). to the family...
We saw the following result in lecture. Suppose that A E Rnn is nonsingular and suppose that r an...
We saw the following result in lecture. Suppose that A E Rnn is nonsingular and suppose that r and r satisfy Az-b and (A+ A)2-b+ b. Let A-A+ A and b-b + b. Finally, assume that メ0 and bメ0. Then SK(A) 11초11 where the norms in 1.1.1) are all mutually consistent. The rest of the proble will form a proof (a) Show that (b) Show that and lell (c) Combine the results of Parts (a) and (1.2.2) to establish (1.1.1)...
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....
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...
Question 3 Consider the ordinary differential equation (ODE) 2xy" + (1 + x)y' + 3y =...
Question 3 Consider the ordinary differential equation (ODE) 2xy" + (1 + x)y' + 3y = 0, in the neighbourhood of the origin. a) Show that x = 0 is a regular singular point of the ODE. (10) b) By seeking an appropriate solution to the ODE, show that G=- (10) i) the roots to the indicial equation of the ODE are 0 and 1/2. [10] ii) the recurrence formula used to determine the power series coefficients, ens when one...
Engineering Mathematics 1 Page 3 of 10 2. Consider the nonhomogeneous ordinary differential equation ry" 2(r...
Engineering Mathematics 1 Page 3 of 10 2. Consider the nonhomogeneous ordinary differential equation ry" 2(r (x - 2)y 1, (2) r> 0. (a) Use the substitution y(x) = u(x)/x to show that the associated homogeneous equation ry" 2(r (x - 2)y 0 transforms into a linear constant-coefficient ODE for u(r) (b) Solve the linear constant-coefficient ODE obtained in Part (a) for u(x). Hence show that yeand y2= are solutions of the associated homogeneous ODE of equation (2). (c) Use...
2. In class we constructed the Green's function for the Laplace operator on the disc with Dirichl...
PDE Greens function:
2. In class we constructed the Green's function for the Laplace operator on the disc with Dirichlet boundary conditions and found that G is given by G(x.xo)-. In (K-xo)-1 In (빻) CU where xXo xol2 Use this Green's function to construct the solution of the equation u(a, θ) = g(θ) and verify the Poisson integral formula (r- |x|) 2π C0 r" 0. ar coS
2. In class we constructed the Green's function for the Laplace operator on...
(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...
plz print ur answer
Question2 Consider the differential equation We saw in class that one solution is the Bessel function (-1)" ( 2n+2 2+n)! 2) n=0 (a) Suppose we have a solution to this ODE in the form y = Σ。:0cmFn+r where 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-must satisfy co c) From the theory about...
1. (a) Write down a second-order differential equation such that y(x) = 3xe-20 is a solution. (b) Determine whether the pair of functions y1 (2) = x sin(x) and y2(x) = cos(x) are linearly inde- pendent. Justify your answer. (c) Write down a second-order ODE which has a solution y(x) that is bounded for all 3 (i.e. y(x) < for all 2) but y(x) is not a constant function.
ODE problem
Implicit equation.2 Consider the ODE (a) Solve the equation by first obtaining explicit equation(s) for y. Here y : R → R is a scalar function. Your answer should be a family of solution parameterized by a single (b) Show that the function y(x) 0 also solves the equation even though it does not belong (c) Sketch a few representative functions of the family found in part (a) together with the parameter (an integration constant). to the family...
We saw the following result in lecture. Suppose that A E Rnn is nonsingular and suppose that r and r satisfy Az-b and (A+ A)2-b+ b. Let A-A+ A and b-b + b. Finally, assume that メ0 and bメ0. Then SK(A) 11초11 where the norms in 1.1.1) are all mutually consistent. The rest of the proble will form a proof (a) Show that (b) Show that and lell (c) Combine the results of Parts (a) and (1.2.2) to establish (1.1.1)...
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....
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...
Question 3 Consider the ordinary differential equation (ODE) 2xy" + (1 + x)y' + 3y = 0, in the neighbourhood of the origin. a) Show that x = 0 is a regular singular point of the ODE. (10) b) By seeking an appropriate solution to the ODE, show that G=- (10) i) the roots to the indicial equation of the ODE are 0 and 1/2. [10] ii) the recurrence formula used to determine the power series coefficients, ens when one...
Engineering Mathematics 1 Page 3 of 10 2. Consider the nonhomogeneous ordinary differential equation ry" 2(r (x - 2)y 1, (2) r> 0. (a) Use the substitution y(x) = u(x)/x to show that the associated homogeneous equation ry" 2(r (x - 2)y 0 transforms into a linear constant-coefficient ODE for u(r) (b) Solve the linear constant-coefficient ODE obtained in Part (a) for u(x). Hence show that yeand y2= are solutions of the associated homogeneous ODE of equation (2). (c) Use...
PDE Greens function:
2. In class we constructed the Green's function for the Laplace operator on the disc with Dirichlet boundary conditions and found that G is given by G(x.xo)-. In (K-xo)-1 In (빻) CU where xXo xol2 Use this Green's function to construct the solution of the equation u(a, θ) = g(θ) and verify the Poisson integral formula (r- |x|) 2π C0 r" 0. ar coS
2. In class we constructed the Green's function for the Laplace operator on...
(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...
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