1.) (10pts) Consider the following differential equation: (x^2)(dy/dx)=2x(sqrt(y))+(x^3)(sqrt(y)) a)Determine whether the equation is separable (S), linear...

1.) (10pts) Consider the following differential equation: (x^2)(dy/dx)=2x(sqrt(y))+(x^3)(sqrt(y))

a)Determine whether the equation is separable (S), linear (L), autonomous (A), or non-linear (N). (An equation could be more than one of these types.)

b)Identify the region of the plane where the Chapter 1 Existence and Uniqueness Theorem guarantees a unique solution exists at an initial condition (x0, y0).

2.(12pts) Consider the IVP: y'+y=y/t , y(2) = 0

For each of the functions y₁(t)and y₂(t) below, decide if it is a solution of the IVP. (Answer is Yes or No, but show, or explain, briefly how you decided for each.)

(a) y₁(t)=te^-1

(b) y₂(t)= t-2

Expert Solution

Colby Messinger answered 2 years ago

Consider the first-order separable differential equation dy/dx = y(y − 1)^2 where the domain of y...

Consider the first-order separable differential equation dy/dx = y(y − 1)^2 where the domain of y ranges over [0, ∞). (a) Using the partial fraction decomposition 1/(y(y − 1)^2) = 1/y −1/(y − 1) +1/((y − 1)^2) find the general solution as an implicit function of y (do not attempt to solve for y itself as a function of x). (b) Draw a phase diagram for (1). Assuming the initial value y(0) =y0, find the interval of values for y0...

dy/dx+(y/2x)=(x/(y^3))

find the differential equation of dy/dx = y+y^3

((sqrtx)+1)dy/dx=(ysqrtx)/(x) + sqrt(y/x), y(2)=1

1. Consider the following second-order differential equation. d^2x/dt^2 + 3 dx/dt + 2x − x^2 =...

1. Consider the following second-order differential equation. d^2x/dt^2 + 3 dx/dt + 2x − x^2 = 0 (a) Convert the equation into a first-order system in terms of x and v, where v = dx/dt. (b) Find all of the equilibrium points of the first-order system. (c) Make an accurate sketch of the direction field of the first-order system. (d) Make an accurate sketch of the phase portrait of the first-order system. (e) Briefly describe the behavior of the first-order...

Consider the curve given by the equation y^2 - 2x^2y = 3 a) Find dy/dx. b)...

Consider the curve given by the equation y^2 - 2x^2y = 3 a) Find dy/dx. b) Write an equation for the line tangent to the curve at the point (1, -1). c) Find the coordinates of all points on the curve at which the line tangent to the curve at that point is horizontal. d) Evaluate d 2y /dx2 at the point (1, -1).

Consider the differential equation dy/dx = y^2 + y - 2 (1) Sketch its phase portrait...

Consider the differential equation dy/dx = y^2 + y - 2 (1) Sketch its phase portrait and classify the critical points. (2) Find the explicit solution of the DE.

For the differential equation dy/dx=sqrt(y^2−36) does the existence/uniqueness theorem guarantee that there is a solution to...

For the differential equation dy/dx=sqrt(y^2−36) does the existence/uniqueness theorem guarantee that there is a solution to this equation through the point 1. (1,6) 2. (4,42) 3. (−2,38) 4. (7,−6)

Find The solution of the Differential Equation of (y+4x+2)dx - dy = 0, y(0) = 3...

Find The solution of the Differential Equation of (y+4x+2)dx - dy = 0, y(0) = 3 ( Please With Steps)

Solve these first-order Differential Equations using an integrating factor. 1. dy/dx+2xy=0 2. dy/dx-y=5 3. dy/dx+y=x 4....

Solve these first-order Differential Equations using an integrating factor. 1. dy/dx+2xy=0 2. dy/dx-y=5 3. dy/dx+y=x 4. (x)dy/dx+(x+1)y=3/x 5. (x^2)dy/dx=e^x-2xy

Question