For each of the following equations sketch the phase portrait of the correspond- ing first order system. Then sketch the graphs of several solutions s(t) for different initial conditions: (a) s" + s = 0 (b) s" s" + s + s = 0 - 8= = 0
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- 6. State whether the equation is ordinary or partial, linear or nonlinear, and give its order. * (x + y) dx + (3x² – 1) dy = 0 ordinary, linear in x, order 1 ordinary, linear in y, order 17 b) For the system, 6 - 1 Y': Y 5 4 Write the natural period of the oscillations, round the answer to three digits after the decimal sign= Find the general solution of the given system. Use a computer system or graphing calculator to construct a direction field and typical solution curves for the system. x' = -4 1 -6 X
- The survival of two species depends on their mutual cooperation. For instance, a species of bee that feeds primarily on the nectar of one plant species and simultaneously pollinates the plant. One simple model of mutualism is given by dx/xt = −ax + bxydy/dt = −my + nxy where a, b, m, and n are all positive constants. Perform a graphical analysis and indicate the trajectory directions in the phase plane.1. (a) Consider the following equation: xy dy – (3x² + 4y²) dx = 0 (i) State the order of the equation, identify whether it's a linear or non-linear equation, and then state the type of equation. (ii) Solve the given equation.A chemistry student heats a beaker that is at room temperature (30 ° C)by inserting it in an oven that has been preheated to 250 °C. If u is the temperature of the beaker in °C, what is the appropriate equation to model the temperature of the beaker. O du/dt = k(u –- 250) O du/dt = k(u – 30) O du/dt = k(250 – u) O du/dt = k(30 – u)
- Q5. Write the governing equation of 1-D heat transfer with convective heat loss and indicate all the parameters involved in it.Match each linear system with one of the phase plane direction fields. (The blue lines are the arrow shafts, and the black dots are the arrow tips.) ? ✓ | 1. z ' = || a' ? 2. ': = ? 3.' = 4. a: = 11 8] -10 3 1 5 -2 1 -5 -13 10] -10 x2 A x2 с x1 (x2 B 2x2/ D Note: To solve this problem, you only need to compute eigenvalues. In fact, it is enough to just compute whether the eigenvalues are real or complex and positive or negative.Sketch the phase portrait for the competing species system: yi = y1(1 – Y1 – 2y2) Y2 = Y2 (1 – Y2 – 2y1) -
- x' = 1 х. Construct the phase plane for the points (2,2), (2,0), (2,-2), (0,2), (0,-2), (-2,2), (-2,0) and (-2,-2). 2.5 1.5 1 0.5 -0.5 -1 -1.5 -2 -2.5 3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2.52a. Find a change of variable that transforms the equation into an autonomous equation change of variable: new equation: b. Sketch the phase line for the resulting equation and use it to sketch graphs of the long-term behaviors of all the qualitatively different solutions for the new variable, and then for the original equation.Match each linear system with one of the phase plane direction fields. (The blue lines are the arrow shafts, and the black dots are the arrow tips.) -0.5 1. i' = -0.5 Yı + Y2 Y2 3. 2y1 4. j' = 0 2 A в D ||| in 2.