(1 point) The functions y = x + are all solutions of equation: xy + 2y...
hellllllllllp please a) Verify that the function y = ?? + is a solution of the differential equation zy' +2y 4x? (x > 0). b) Find the value ofe for which the solution satisfies the initial condition (2) - 5. = Submit Question a) Verify that the function y=x? + с 2 is a solution of the differential equation ry' + 2y = 4x², (x > 0). b) Find the value of c for which the solution satisfies the initial...
1. Consider the equation xy" - 2y' + (2 - x)y = 0,x > 0. We can easily verify that y(x) = e* is a solution of the equation. Use reduction of order to determine the general solution of the equation.
Find the function y=y(x) (for x>0) which satisfies the separable differential equation dy/dx = (4+17x)/(xy^2). ;x>0 with the initial condition: y(1)=2
3.1: Characteristic polynomial, linear independence 8. Consider the following differential equation: xy" - (x + 1)y + y = 0 The functions, Y1 = +1 72 = 4x + 4 are solutions to this differential equation. Find the general solution, or explain why we don't have enough information to do so.
Suppose that all solutions of the differential equation f(x,y), y = g(x,y) exist for all time and that f and g are smooth (Co) functions. Let 7(t) be the solution of the initial value problem with γ(0) = (1,2). Prove or give a counterexample to the statement that the w-limit set of γ can contain more than one critical point. Suppose that all solutions of the differential equation f(x,y), y = g(x,y) exist for all time and that f and...
(1 point) A Bernoulli differential equation is one of the form dy dc + P(x)y= Q(x)y" Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = yl-n transforms the Bernoulli equation into the linear equation du dr +(1 – n)P(x)u = (1 - nQ(x). Consider the initial value problem xy + y = 3xy’, y(1) = -8. (a) This differential equation can be written in the form (*)...
(1 point) A first order linear equation in the form y' + p(x)y = f(x) can be solved by finding an integrating factor u(x) = expl (1) Given the equation xy' + (1 +4x) y = 10xe 4* find y(x) = (2) Then find an explicit general solution with arbitrary constant C. y = (3) Then solve the initial value problem with y(1) = e-4 y =
1 point) Given that y -x is a solution of dy dx2 in x > 0, find another solution yc of ус the same equation such that (xy.) is a fundamental set of solutions
(1 point) A first order linear equation in the form y +p(x)y -f(x) can be solved by finding an integrating factor H(x)exp /p(x) dx (1) Given the equation xy + (1 + 4x) y-6xe_4x find (x)-| xeN4x) (2) Then find an explicit general solution with arbitrary constant C (3) Then solve the initial value problem with y(1)e
(1 point) Let x and y have joint density function p(2, y) = {(+ 2y) for 0 < x < 1,0<y<1, otherwise. Find the probability that (a) < > 1/4 probability = (b) x < +y probability =