If y is a known nonvanishing solution of y" p(t)y + q(t)y 0, then a second...
Bonus (Abel's formula) a) Show that if y1 and y2 are solutions to the differential equation y"p(t)y(t)y 0 where p and q are continuous on an interval I, then the Wronskian of y and y2, W(y1,y2) (t) is given by - Sp(t)dt ce W(y1, y2)(t) where c depends on y and y2 (b) Use Abel's formula to find the Wronskian of two solutions to the differential equation ty"(t 1)y 3y 0 Do not solve the differential equation
find Y1=, Y2=, and W(t)= (1 point) Find the function yi of t which is the solution of 25y" – 40y' + 12y = 0 y(0) = 1, yf(0) = 0. with initial conditions Yi = Find the function y2 of t which is the solution of 25y" – 40y' + 12y = 0 with initial conditions Y2 = Find the Wronskian W(t) = W(y1, y2). W(t) = Remark: You can find W by direct computation and use Abel's theorem...
Consider the differential equation, L[y] = y'' + p(t)y' + q(t)y = 0, (1) whose coefficients p and q are continuous on some open interval I. Choose some point t0 in I. Let y1 be the solution of equation (1) that also satisfies the initial conditions y(t0) = 1, y'(t0) = 0, and let y2 be the solution of equation (1) that satisfies the initial conditions y(t0) = 0, y'(t0) = 1. Then y1 and y2 form a fundamental set...
Consider the ordinary differential equation: t2y" + 3ty' +y = 0. 1 (3 points) e) Use Abel's formula to find the Wronskian of any two solutions of this equation and W[y1,y2](t). What do you observe? compare it to = t1 and y2(t) = t-1 nt represent a fundamental set of solu f) (2 points) Determine if y1 (t) tions (2 points) Find the general solution of t2y" +3ty' +y = 0. g) Solve the initial value problem t2y" + 3ty/...
Given a second order linear homogeneous differential equation а2(х)у" + а (х)У + аo(х)у — 0 we know that a fundamental set for this ODE consists of a pair linearly independent solutions yı, V2. But there are times when only one function, call it y, is available and we would like to find a second linearly independent solution. We can find y2 using the method of reduction of order. First, under the necessary assumption the a2(x) F 0 we rewrite...
pls do all questions. thanx 1. [5 Consider the IVP rty(t) + 2 sin(t)y(t) = tan(t) y(5)=2 Does a unique solution of the IVP exist? Do not solve the IVP but fully justify you answer. What is the IOE? 2. 4 Consider the ODE Using undetermined coefficients, what is an approprite guess for the coefficient (s) in yp but fully justify you answer. ? Do not solve for 3. [10] Solve the IVP. Use any approach you like y(x) 6y'(x)...
2. (3+4+4+4 pts) In this problem, we discuss a method of solving SOL equations known as Reduction of Order. Given an equation y" +p(a)y' +9(2)y = 0, and assuming yi is a solution, Reduction of Order asks: does there exist a second, linearly-independent solution y2 of the form y2 = u(x)41 for some function u(x)? See Section 3.2, Exercise 36 for reference). We'll now use this to solve the following problem. (a) Consider the SOL differential equation sin(x)y" — 2...
Empty Part only Let L[y]: y"" y'+4xy, yi (x): = sinx, y2(x): =x. Verify that L[y11(x) 4xsinx and to the following differential equations. Ly2 (X)= 4x1. Then use the superposition principle (linearity) to find a solution (a) Lly] 8x sin x - 4x2-1 (b) Lly] 16x+4 -24x sin x y1(x)- cos x tlV]¢»= 4x° Substituting yi (x), y, '(x), and y"(x) into L[y] y""+y' +4xy yields Lfy1(x) 4xsinx. Now verify that +1. Calculate y2'(x) y2'(x) 1 Calculate y2"(x). У2"(х)%3D 0...
Find the solution y of the initial value problem 3"(t) = 2 (3(t). y(1) = 0, y' (1) = 1. +3 g(t) = M Solve the initial value problem g(t) g” (t) + 50g (+)? = 0, y(0) = 1, y'(0) = 7. g(t) = Σ Use the reduction order method to find a second solution ya to the differential equation ty" + 12ty' +28 y = 0. knowing that the function yı(t) = + 4 is solution to that...
2. Given two initial value problems, у" — р(г)у +q()у +r(x) with a <I<b,y(a) — с,1 (а) —0 (1) and у" — р(г)у + g(х)у with a < r <ь,y(a) — 0, and / (а) — 1 (2) [a, b) where p(x), q(z) and r(x) Show that given two solutions yı(x), y2(x) to the linear value problems above, (1) and (2), respectively, then there exists a solution y(x) to a linear boundary value problem above where y(a) %3D 0, у...