Chapter 1: Q5.4-2E (page 1)

In Problems 1 and 2, verify that the pair x(t), and y(t) is a solution to the given system. Sketch the trajectory of the given solution in the phase plane.
$\frac{dx}{dt} = 1, \frac{dy}{dt} = 3 x^{2}; x (t) = t + 1, y (t) = t^{3} + 3 t^{2} + 3 t$

Short Answer

Expert verified

Hence, it is verified that $x (t), y (t)$ is a solution to the given system.

Step by step solution

Get the result in form of x and y

Here the system is;

$\frac{dx}{dt} = 1 \frac{dy}{dt} = 3 x^{2}$

And

$x (t) = t + 1 y (t) = t^{3} + 3 t^{2} + 3 t$

Then

$\frac{dx}{dt} = 1 \frac{dy}{dt} = 3 (t^{2} + 2 t + 1) = 3 (t + 1)^{2} = 3 (x (t))^{2}$

Therefore, the pair (x(t), y(t)) is a solution to the system.

Also,

$y (t) + 1 = t^{3} + 3 t^{2} + 3 t + 1 = (t + 1)^{3} = (x (t))^{3} y = x^{3} - 1$

Get the result

Since $\frac{dx}{dt} = 1$ x is increasing,the flow is from left to right along the curve.

Sketch

This is the required result.

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In Problems 1 and 2, verify that the pair x(t), and y(t) is a solution to the given system. Sketch the trajectory of the given solution in the phase plane.
$\frac{dx}{dt} = 1, \frac{dy}{dt} = 3 x^{2}; x (t) = t + 1, y (t) = t^{3} + 3 t^{2} + 3 t$

Short Answer

Step by step solution

Get the result in form of x and y

Get the result

Sketch

One App. One Place for Learning.

Most popular questions from this chapter

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In Problems 1 and 2, verify that the pair x(t), and y(t) is a solution to the given system. Sketch the trajectory of the given solution in the phase plane.dxdt=1,dydt=3x2;x(t)=t+1,y(t)=t3+3t2+3t

Short Answer

Step by step solution

Get the result in form of x and y

Get the result

Sketch

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Geometry

Theoretical and Mathematical Physics

Mechanics Maths

Statistics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Problems 1 and 2, verify that the pair x(t), and y(t) is a solution to the given system. Sketch the trajectory of the given solution in the phase plane.
$\frac{dx}{dt} = 1, \frac{dy}{dt} = 3 x^{2}; x (t) = t + 1, y (t) = t^{3} + 3 t^{2} + 3 t$