4. (a) Find a second order linear equation which has y as two of its solutions....
It's in Mathematical Methods for Physicists 7e, Arfken ch7.7 Inhomogeeous linear ODEs. Please help. Thank you. 7.7.1 If our linear, second-order ODE is inhomogeneous, tha is, of the form of Eq. (7.94), the most general solution is where yi and y2 are independent solutions of the homogeneous equation Show that yi(x)2()Fsds W[yi(s), y2()) Wyi(), y2(s) with Wlyxy2x)) the Wronskian of yi(s) and y2(s) Find the general solutions to the following inhomogeneous ODEs: 7.7.1 If our linear, second-order ODE is inhomogeneous,...
2. You can use Dand write an operator instead of an equation in this question. (a) Find a constant coefficient linear homogeneous differential equation of lowest order that has n(x)-x , y2(z) = x2 , and y3(z) = eェamong its solutions. (b) Now find a different linear homogeneous differential equation of an order lower than the one in (a) that has the same y1,U2,U3 among its solutions. (c) Find a constant coefficient linear homogeneous differential equation of lowest order that...
Find a second order linear equation L(y) = f(t) with constant coefficients whose general solution is: @ y=Cje24 + C261 + te3t @ (a) The solution contains three parts, so it must come from a nonhomogeneous equation. Using the two terms with undefined constant coefficients, find the characteristic equation for the homogeneous equation. (b) Using the characteristic equation find the homogeneous differential equation. This should be the L(y) we're looking for. (c) Since we have used two terms from the...
2. (Undetermined Coefficients... In Reverse) Find a second order linear equation L(y) = f(0) with constant coefficients whose general solution is: y=C et + Cell + tet (a) The solution contains three parts, so it must come from a nonhomogeneous equation. Using the two terms with undefined constant coefficients, find the characteristic equation for the homogeneous equation (h) Using the characteristic equation find the homogeneous differential equation. This should be the L(y) we're looking for. (c) Since we have used...
Find a second order homogeneous linear differential equation whose general solution is A tan x + B sin x (A, B constant). [Hint: Use the fact that tan x and sin x are, individually, solutions and solve for the coefficients in standard form.]
2. a) (7 pnts) Solve the second order homogeneous linear differential equation y" - y = 0. b) (6 pnts) Without any solving, explain how would you change the above differential equation so that the general solution to the homogeneous equation will become c cos x + C sinx. c) (7 pnts) Solve the second order linear differential equation y" - y = 3e2x by using Variation of Parameters. 5. a) (7 pnts) Determine the general solution to the system...
You are told that a certain second order, linear, constant coefficient, homogeneous ode has the solutions y1(x) = e^γx cos ωx, and y2(x) = e^γx sin ωx, where γ and ω are real-valued parameters and −∞ < x < ∞. 4. You are told that a certain second order, linear, constant coefficient, homogeneous ODE has the solutions where γ and w are real-valued parameters and-oo < x < oo. (a) Compute the Wronskian for this set of solutions. (b) Using...
3. Given the second-order equation y" +3y2y sin 3t a) Perform an a priori analysis and predict what the long-term behavior of the solutions will be (briefly explain b) Use complexification to find its general solution (or, by loosing 8 points, do it without complexifying) c) Check that the predicted behavior matches the analytic solution found 3. Given the second-order equation y" +3y2y sin 3t a) Perform an a priori analysis and predict what the long-term behavior of the solutions...
Given that 6e22 and 5e 3T are solutions of a second order linear homogeneous differential equation with constant coefficients, find this differential equation. a) y" – 1ly' + 30y = 0 b) c) d) e) y" + 1ly' + 30y = 0 y" – y' – 6y = 0 y" + 1ly' – 30y = 0 y" + y' - 6y=0 f) None of the above.
Problem 1 (14 points) (a) Find the general solution to a third-order linear homogeneous differential equation for y(1) with real numbers as coefficients if two linearly independent solutions are known to be e-21 and sin(3.c). e (b) Determine that differential equation described in part (a).