first diff eq test

Determine the order of the given differential equation; also state whether the equation is linear or nonlinear.t2 d^2y/dt^2 + tdy/dt + 2y = sin t : 1.3, 1

Determine the order of the given differential equation; also state whether the equation is linear or nonlinear.d^4y/dt^4 + d^3y/dt^3 + d^2y/dt^2 + dy/dt + y =1 : 1.3, #3

verify that each given function is a solution of the differential equation.y'' + y = sec t, 0< t < π/2; y = (cos t) ln cos t + t sin t : 1.3, #13

determine the values of r for which the given differentialequation has solutions of the form y = e^rty'' + y' − 6y = 0 : 1.3#17

solve the given differential equationy' = x^2/y : 2.2, #1

solve the given differential equationy' + y^2 (sin x) =0 : 2..2, #3

determine (without solving the problem) an interval in which the solution of the given initial value problem is certain to exist(4 − t^2)y' + 2ty = 3t^2, y(1) = −3 : 2.4, #5

state where in the ty-plane the hypotheses ofTheorem 2.4.2 are satisfied.y' = (t − y)/(2t + 5y) : 2.4, #7

Determine whether each of the equations is exact. if so, find the solution((ye^(xy))cos 2x − 2e^(xy) sin2x + 2x) + (xe^(xy) cos 2x − 3)y'= 0 : 2.6, #9

Determine whether each of the equations is exact. if so, find the solution(x ln y + xy) + (y ln x + xy)y' = 0; x > 0, y > 0 : 2.6, #11

determine whether the equation is exact. if not, determine an integrating factor and solve the equatione^x + (e^x *cot y + 2y csc y)y' = 0 : 2.6, #29

find the general solution of the given differential equationy'' + 2y' − 3y =0 : 3.1, #1

find the solution of the given initial value problem6y'' − 5y' + y = 0,y(0) = 4, y'(0) = 0 : 3.1, #11

find the solution of the given initial value problem. describe its behavior as t increasesy'' + 3y' = 0, y(0) = −2, y'(0) = 3 : 3.1, #12

verify that the functions y1 and y2 are solutions of the given differential equation. Do they constitute a fundamental set of solutions?y'' − 2y' + y = 0; y1(t) = e^t , y2(t) = te^t : 3.2, #25

verify that the functions y1 and y2 are solutions of the given differential equation. Do they constitute a fundamental set of solutions?(1 − x cot x)y'' − xy' + y = 0, 0 < x < π;y1(x) = x, y2(x) = sin x : 3.2, #27