Find and then solve the Euler-Lagrange equations satisfied by minimizers yo of dx. .b (c) Ily(-)]-/ y(x) (y,(x) + xy(x)) dx. Find and then solve the Euler-Lagrange equations satisfied by minimiz...
Problem 1. For each of the following functions f (x,y,y'), use the Euler-Lagrange equations to derive a differential equation for the function y(x) that minimizes the functional Fy (x,y,y') dx. Do all calculations by hand. 1. f(x,y,y') = { (y')? – eXy 2. f (x, y, y') = 3y2 – ery 3. f (x,y, y') =y(1+(y)2) "? 4. f (x,y,y') =
(a) Show that, if y satisfies the Euler-Lagrange equation associated with the integral 2. qy2) dx, I = (6) where p() and q(x) are known functions, then I has the value 12 (b) Show that, if y satisfies the Euler-Lagrange equation associated with (6) and if z(x) is an arbitrary differentiable function for which z(x)z(r2) = 0, (7) 1 then (yyz)da= 0. + Hence show that by replacing y in (6) by the function (y + z), where the condition...
Question 2: solve the differential equations a) (xy - y)dx + - x)dy = 0
Solve the Cauchy-Euler differential equations: x^2d^2y/dx^2 – 12xdy/dx + 81y = 0
Solve the Euler ODE 7. x?y"+ xy'– 9y = x3
Find the Euler-Lagrange Equations for Е (и, ик) = a) + их х
solve the following differential equations (e* + 2y)dx + (2x – sin y)dy = 0 xy' + y = y? (6xy + cos2x)dx +(9x?y? +e")dy = 0 +2ye * )dx = (w*e * -2rcos x) di
3. Find all critical points of dt dt with the constraint PP = 8 0 (c and boundary conditions x(0) - 0, x(1)- 3. Hint: Write the Euler Lagrange equation (there is no dependence on t), and then use the boundary conditions and the constraint to reach a system of 2 equations (with quadratic terms) of two unknown constants a, b Solve it by first finding a quadratic equation for a/b 3. Find all critical points of dt dt with...
using delta(pi) method Find the Euler-Lagrange Equations for the functional (using δπ) 5. EI 0 for constants, E, I and P Find the Euler-Lagrange Equations for the functional (using δπ) 5. EI 0 for constants, E, I and P
3) (xy + y) dx + (x + x²y + x²g?)dy=0 differential equations