Question 5: Show that the nontrivial solution of the equation xy" +y+y=0 Satisfying the conditions y'(1) = 0, y' (e) = 0 form an orthogonal set on the indicated interval. Solve the equation, impose the given boundary conditions, obtain the characteristic funagons, and verify their orthogonality by direct integration

Which of the following equations are homogenous or not : x³+y3 1. у' ху? 2. y' 3D еу + sin: y 3. dy =dx x+y 4. y' = x2 x+y 5. y' = X-y х-у+3 6. y' = х+у-1

the linearization of (fi,f2) = (x²y+ 3xy², 3x³ + 2xy) at (1,0) to approximate the value at (0.9,-0.1).

1. Provide at least 6 vectors of the slope field of the differential equation y' = x − y and use them to sketch the solution curve through the point (1,2).

Find the velocity and acceleration vectors and the equation of the tangent line for the curve r(t) = √3ti + e¹¹j + 6e¯¹k at t = 0. (Use symbolic notation and fractions where needed. Give your answers in the form (*,*,*).) v(0) = a(0) = (Use symbolic notation and fractions where needed. Give your answers in the form (*,*,*). Use t for the parameter that takes all real values.) 1(t)

Sketch the direction field of the DE y'=2x+y.  (Plot 9 or more points) Then sketch approximate solution that passes through y(1)=1

a.) Form the complimentary solution to the homogeneous equation. y_c(t) = c_1 [ _ _] + c_2 [ _ _] b.) Construct a particular solution by assuming the form y_P(t) = e^(-3t)a and solving for the undetermined constant vector a. y_P(t) = [ _ _] c.) Form the general solution y(t) = y_c(t) + y_P(t) and impose the initial condition to obtain the solution of the initial value problem. [y_1(t) y_2(t)] = [ _ _]

Write parametric equations for a(t) at (1,1) a(t) = <1, t, 1/(e+t^3)>

Consider the differential equation x+y-1. a) Explain why you cannot solve this differential equation by separation of variables. b) Shown below is a slope-field for the given differential equation. Sketch the particular solutions that pass through the points (2. --1) and (3,3) c) Find in terms of x and y. Describe the region in the xy-plane in which all solution curves to the differential equation are concave up. d) Let y = f(x) be a particular solution to the differential equation with initial condition f(0) = 1. Does f has a relative min, max, or neither at x= 0? Justify your answer. %3D

Find the direction field for the equation: y' = x + sin(y) + cos (x) at the point A = (0, pi3)