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1. We say p (ro. yo, 20) is a regular point for the equation F(x, y,) 0 if the equation either de...
a) Show that the equation 23 : 1 f(z, Y, z) := +y+ defines a smooth...
a) Show that the equation 23 : 1 f(z, Y, z) := +y+ defines a smooth surface S. b) Show that for any (r, y, z) E S, the gradient vector (fz(x, y, z), fy(, y, z), f:(x, y, z)) of f is a normal vector to S. (Hint: let a = x(t), y = y(t), z = z(t) be a curve in the surface passing through a point (o, Yo, 2o) in S, where ro = r(0), yo: y(0),...
7.7.4 The hypotheses of Theorem 7.24 require that f be differentiable on all of the interval I. Y...
7.7.4 The hypotheses of Theorem 7.24 require that f be differentiable on all of the interval I. You might think that a positive derivative at a single point also implies that the function is increasing, at least in a neighborhood of that point. This is not true. Consider the function /(z) _{0,/2 + ra sin.ri. if 0 (e) Prove that if a function F is differentiable on a neighborhood of ro with F(ro)0 and F is continuous at zo, then...
Co are 5. Suppose that the functions f :R3 R, g:R R, and h:RR ously differentiable and let (xo. o...
co are 5. Suppose that the functions f :R3 R, g:R R, and h:RR ously differentiable and let (xo. o, zo) be a point in R3 at which f(xo, yo, zo-g(xo, yo, zo)sh(xo, yo, zo)s0 and By considering the set of solutions of this system as consisting of the intersection of a surface with a path, explain why that in a neighborhood of the point (xo, yo, Zo) the system of equations f(x, y, z) g(x, y, 2)0 hCx, y,...
(d) Show that the point ro 0 is a regular singular point for the equation and that the solution about the point 20 0 can be written as where x>0 (2n-1)]], [n!1.3.5 , Lnsl (d) Show that th...
(d) Show that the point ro 0 is a regular singular point for the equation and that the solution about the point 20 0 can be written as where x>0 (2n-1)]], [n!1.3.5 , Lnsl
(d) Show that the point ro 0 is a regular singular point for the equation and that the solution about the point 20 0 can be written as where x>0 (2n-1)]], [n!1.3.5 , Lnsl
1) Show that zo-0 is a regular singular point for the diferenta equation Zo = 0 is a regular sing...
please show the recurrence formula
1) Show that zo-0 is a regular singular point for the diferenta equation Zo = 0 is a regular singular point for the differential equation 15ェy" + (7 + 15r)y, +-y = 0, x>0. Use the method of Frobenius to obtain two linearly independent series solutions about zo Find the radii of convergence for these series. Form the general solution on (0, 0o). 0.
1) Show that zo-0 is a regular singular point for the...
Problem 1 Let gi(x, y, z)-y, 92(x, y, z)z and f(x, y, z) is a differential function We introduce ...
Problem 1 Let gi(x, y, z)-y, 92(x, y, z)z and f(x, y, z) is a differential function We introduce F(x, y, z, A, )-f(x, y, z) - Xgi(x, y, z) - Hg2(x, y, 2). ·Show that the Lagrange system for the critical points off with constraints gi (x, y, z) = 92(x,y, z)0: F(zo, yo, 20, λο, μο)-(0, 0, 0, 0, 0) is equivalent to the one-dimensional critical point equation: df dr(ro, 0, 0) = 0, 30 = 20 =...
4. Consider the equation zy" - 2y' y 0 (a) Explain why r 0 is a regular singular point for the given equation (b) Let ri >r2 be two indical roots of the given equation. Using Frobenius'...
4. Consider the equation zy" - 2y' y 0 (a) Explain why r 0 is a regular singular point for the given equation (b) Let ri >r2 be two indical roots of the given equation. Using Frobenius' method, find a series solution n(x)-z"Ση_0Cnz". (c) Find the second solution of the form Σ000 bnXntr2 with boメ0, or i (z) Inr +bn+r2 with the first three nonzero terms of the series with coefficient bn
4. Consider the equation zy" - 2y' y...
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....
4. Suppose you are given an equation of the form F(x, y,z) 0. Then we can say that each of the variables is defined implicitly as a function of the others. 2 a) If F and z(x, y) are both assumed to b...
4. Suppose you are given an equation of the form F(x, y,z) 0. Then we can say that each of the variables is defined implicitly as a function of the others. 2 a) If F and z(x, y) are both assumed to be differentiable, fnd in terms of partial derivatives of F. b) Under similar assumptions on the other variables, find
4. Suppose you are given an equation of the form F(x, y,z) 0. Then we can say that each...
Consider the equation 3x²y" + x(2 – xy + xy = 0 with regular singular point...
Consider the equation 3x²y" + x(2 – xy + xy = 0 with regular singular point Xo = 0. (a) Find the indicial roots ri, r2, with ri r2. Show your calculations. (b) Which of the following is true for the equation above: Indicate the letter of your choice and explain your choice. % There are two linearly independent convergent series solutions of the form yı (x) = x Š cux" and y(x) = x Š b,x". H0 N=0 (1)...
a) Show that the equation 23 : 1 f(z, Y, z) := +y+ defines a smooth surface S. b) Show that for any (r, y, z) E S, the gradient vector (fz(x, y, z), fy(, y, z), f:(x, y, z)) of f is a normal vector to S. (Hint: let a = x(t), y = y(t), z = z(t) be a curve in the surface passing through a point (o, Yo, 2o) in S, where ro = r(0), yo: y(0),...
7.7.4 The hypotheses of Theorem 7.24 require that f be differentiable on all of the interval I. You might think that a positive derivative at a single point also implies that the function is increasing, at least in a neighborhood of that point. This is not true. Consider the function /(z) _{0,/2 + ra sin.ri. if 0 (e) Prove that if a function F is differentiable on a neighborhood of ro with F(ro)0 and F is continuous at zo, then...
co are 5. Suppose that the functions f :R3 R, g:R R, and h:RR ously differentiable and let (xo. o, zo) be a point in R3 at which f(xo, yo, zo-g(xo, yo, zo)sh(xo, yo, zo)s0 and By considering the set of solutions of this system as consisting of the intersection of a surface with a path, explain why that in a neighborhood of the point (xo, yo, Zo) the system of equations f(x, y, z) g(x, y, 2)0 hCx, y,...
(d) Show that the point ro 0 is a regular singular point for the equation and that the solution about the point 20 0 can be written as where x>0 (2n-1)]], [n!1.3.5 , Lnsl
(d) Show that the point ro 0 is a regular singular point for the equation and that the solution about the point 20 0 can be written as where x>0 (2n-1)]], [n!1.3.5 , Lnsl
please show the recurrence formula
1) Show that zo-0 is a regular singular point for the diferenta equation Zo = 0 is a regular singular point for the differential equation 15ェy" + (7 + 15r)y, +-y = 0, x>0. Use the method of Frobenius to obtain two linearly independent series solutions about zo Find the radii of convergence for these series. Form the general solution on (0, 0o). 0.
1) Show that zo-0 is a regular singular point for the...
Problem 1 Let gi(x, y, z)-y, 92(x, y, z)z and f(x, y, z) is a differential function We introduce F(x, y, z, A, )-f(x, y, z) - Xgi(x, y, z) - Hg2(x, y, 2). ·Show that the Lagrange system for the critical points off with constraints gi (x, y, z) = 92(x,y, z)0: F(zo, yo, 20, λο, μο)-(0, 0, 0, 0, 0) is equivalent to the one-dimensional critical point equation: df dr(ro, 0, 0) = 0, 30 = 20 =...
4. Consider the equation zy" - 2y' y 0 (a) Explain why r 0 is a regular singular point for the given equation (b) Let ri >r2 be two indical roots of the given equation. Using Frobenius' method, find a series solution n(x)-z"Ση_0Cnz". (c) Find the second solution of the form Σ000 bnXntr2 with boメ0, or i (z) Inr +bn+r2 with the first three nonzero terms of the series with coefficient bn
4. Consider the equation zy" - 2y' y...
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....
4. Suppose you are given an equation of the form F(x, y,z) 0. Then we can say that each of the variables is defined implicitly as a function of the others. 2 a) If F and z(x, y) are both assumed to be differentiable, fnd in terms of partial derivatives of F. b) Under similar assumptions on the other variables, find
4. Suppose you are given an equation of the form F(x, y,z) 0. Then we can say that each...
Consider the equation 3x²y" + x(2 – xy + xy = 0 with regular singular point Xo = 0. (a) Find the indicial roots ri, r2, with ri r2. Show your calculations. (b) Which of the following is true for the equation above: Indicate the letter of your choice and explain your choice. % There are two linearly independent convergent series solutions of the form yı (x) = x Š cux" and y(x) = x Š b,x". H0 N=0 (1)...
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