Chapter 1.1, Problem 33P

Lemmas let x--Ax, y = By where As P 0 (a) B. ( &B to then the linen system x = Ax, y = By are linearly equivalent iff Bad

No chatgpt pls will upvote Already got wrong chatgpt answer

Not use ai please

No chatgpt pls will upvote Already got wrong chatgpt answer Plz

match the equation to it's respective directional field in the image, justify your answer   a. dy/dx=x-1 b. dy/dx=1 - y^2 c. dy/dx=y^2  - x^2 d. dy/dx=1-x e. dy/dx=1-y f. dy/dx=x^2  - y^2 g. dy/dx=1+y h. dy/dx=y^2  - 1

4. The runway at the Piarco International airport has an equation of -3(x-2y) = 6. If the Priority Bus Route passes through the geometric coordinate (1,-9) and is perpendicular to the runway at the Piarco International airport. Determine the following: a. State two geometric coordinates which the runway at the Piarco International airport passes through. b. Derive the equation of the Priority Bus Route. [2 marks] [6 marks]

Q4*) Find the extremals y, z of the the functional I = 1 (2yz - 2x² + y²² 12 - 212) dx, with y(0) = 0, y(1) = 1, z(0) = 0, ≈(1) = 0.

Solve the following initial value problem over the interval from t= 0 to 2 where y(0)=1. dy yt² - 1.1y dt Using Euler's method with h=0.5 and 0.25.

Q5*) Write down an immediate first integral for the Euler-Lagrange equation for the integral I = = F(x, y, y″) dx. Hence write down a first integral of the Euler-Lagrange equation for the integral I 1 = √(xy ² + x³y²) dx. Find the general solution of this ordinary differential equation, seeking first the complementary function and then the particular integral. (Hint: the ODE is of homogeneous degree. And, for the particular integral, try functions proportional to log x.)