STM 2100 Quiz April 2 2020 Consider the radioactive decay model dy/dt = -0.1y y(0) =...

STM 2100 Quiz April 2 2020

Consider the radioactive decay model

dy/dt = -0.1y

y(0) = 10

a. Use separation of variables to solve the ODE

b. At what time does y(t) = 5?

2. Consider the logistic model given by the following ODE

dy/dt = y(1-y)

y(0) = 0.1

a. Use separation of variables to rewrite the problem as an equality of two integrals

b. Use partial fractions to rewrite 1/y(1-y) as the sum of two fractions, each of which has a single factor in its denominator

c. Substitute your result for (b) into (a), and solve for y(t)

d. At what time does y(t) = 0.2? At what time does y(t) = 0.4?

Expert Solution

if satisfied with the explanation, please rate it up..

milcah answered 3 years ago

Use a LaPlace transform to solve d^2x/dt^2+dx/dt+dy/dt=0 d^2y/dt^2+dy/dt-4dy/dt=0 x(0)=1,x'(0)=0 y(0)=-1,y'(0)=5

Consider the following initial value problem dy/dt = 3 − 2t − 0.5y, y (0) =...

Consider the following initial value problem dy/dt = 3 − 2*t − 0.5*y, y (0) = 1 We would like to find an approximation solution with the step size h = 0.05. What is the approximation of y(0.1)?

Consider the system modeled by the differential equation dy/dt - y = t with initial condition y(0) = 1

Consider the system modeled by the differential equation dy/dt - y = t with initial condition y(0) = 1 the exact solution is given by y(t) = 2et − t − 1 Note, the differential equation dy/dt - y =t can be written as dy/dt = t + y using Euler’s approximation of dy/dt = (y(t + Dt) – y(t))/ Dt (y(t + Dt) – y(t))/ Dt = (t + y) y(t + Dt) =...

dx dt =ax+by dy dt =−x − y, 2. As the values of a and b...

dx dt =ax+by dy dt =−x − y, 2. As the values of a and b are changed so that the point (a,b) moves from one region to another, the type of the linear system changes, that is, a bifurcation occurs. Which of these bifurcations is important for the long-term behavior of solutions? Which of these bifurcations corresponds to a dramatic change in the phase plane or the x(t)and y(t)-graphs?

3. Consider the IVP: dy =ty^1/3; y(0)=0,t≥0. dt Both y(t) = 0, (the equilibrium solution) and...

3. Consider the IVP: dy =ty^1/3; y(0)=0,t≥0. dt Both y(t) = 0, (the equilibrium solution) and y(t) = ?(1/3t^2?)^3/2 are solutions to this IVP. (a) Show that the trivial solution satisfies the IVP by first verifying that it satisfies the initial condition and then verifying that it satisfies the differential equation. (b) Show that the other solution satisfies the IVP again by first verifying it satisfies the initial condition and then verifying that it satisfies the differential equation. (c) Explain...

Consider the non linear ODE: (dx/dt) = -y = f(x,y) (dy/dt) = x^2-x = g(x,y) (a)....

Consider the non linear ODE: (dx/dt) = -y = f(x,y) (dy/dt) = x^2-x = g(x,y) (a). Compute all critical points (b) Derive the Jacobian matrix (c). Find the Jacobians for each critical point (d). Find the eigenvalues for each Jacobian matrix (e). Find the linearized solutions in the neighborhood of each critical point (f) Classify each critical point and discuss their stability (g) Sketch the local solution trajectories in the neighborhood of each critical point

6. Consider the initial value problem dy/ dt = t ^2 y , y(1) = 1...

6. Consider the initial value problem dy/ dt = t ^2 y , y(1) = 1 . (a) Use Euler’s method (by hand) to approximate the solution y(t) at t = 2 using ∆t = 1, ∆t = 1/2 and ∆t = 1/4 (use a calculator to approximate the answer for the smallest ∆t). Report your results in a table listing ∆t in one column, and the corresponding approximation of y(2) in the other. (b) The direction field of dy/dt...

Solve the given initial-value problem. dx/dt = y − 1 dy/dt = −6x + 2y x(0)...

Solve the given initial-value problem. dx/dt = y − 1 dy/dt = −6x + 2y x(0) = 0, y(0) = 0

Consider the differential equation dy/dt = 2?square root(absolute value of y) with initial condition y(t0)=y0 •...

Consider the differential equation dy/dt = 2?square root(absolute value of y) with initial condition y(t0)=y0 • For what values of y0 does the Existence Theorem apply? • For what values of y0 does the Uniqueness theorem apply? • Verify that y1(t) = 0 solves the initial value problem with y0 = 0 • Verify that y2(t) = t2 solves the initial value problem with y0 = 0 • Does this violate the theorems from this section 1.5? Why or why...

Solve the following initial value problems (1) dy/dt = t + y y(0) = 1 so...

Solve the following initial value problems (1) dy/dt = t + y y(0) = 1 so y(t) = (2) dy/dt = ty y(0) = 1 so y(t) =

Question