Problem 1. Find all equilibrium solutions of the following system of differential equa- (a) tions: -1 #[0]-[RB][B][2] d dt = X + y² (b) Find three Picard's iterations of IVP x(0) = 2, y(0) = 5. (c) Plot the vector field of this system on [−7, 7]².
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- +(?) v(e) = 2. Differential equation for the circuit is Lv" (t) + Rv'(t) + = 0. R Voltage source L Let V be the vector space of all solutions of the above in the form v(t) = e-bt (cı + c2t), here v(t) is the voltage across the capacitor. Find a basis for V.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.1. Let x' = ·[₁ ³] -3 -2 6 X. (a) Find the general solution of the given system of equations. (b) Draw a directional field and a few of the trajectories.
- This is the first part of a three-part problem. Consider the system of differential equations y1 - 2y2, y1 + 4y2. Yi Y2 Rewrite the equations in vector form as '(t) = Ay(t). j'(t) = 身(2) Match equations a) y' x-y %3D and b) y' = - with the direction fields (A) and (B). 10. 1+x2 (A) (B) 02 43.) This is differential Calculus subject. A PARTICLE IS MOVING ALONG A HORIZONTAL LINE ACCORDING TO THE GIVEN EQUATION.The Equation is S = t^3 - 9t^2 + 15t ; t>=0.where (s) meters is the directed distance of the particle from the originat (t) seconds. where (v) meters per second is the instantaneous acceleration of the particle. 1. FIND (v) AND (a) IN TERMS OF (t).2. MAKE A TABLE THAT GIVES A Description of THE POSITION AND MOTION of the particle, include in the table the intervals of time when the PARTICLE IS MOVING TO THE LEFT AND RIGHT include in the table when the VELOCITY IS INCREASING AND DECREASING include in the table when the SPEED IS INCREASING AND DECREASING include in the table the POSITION OF THE PARTICLE with respect to the origin during these intervals of time. and Show the BEHAVIOR OF THE MOTION
- (2) Consider the following vector ODE Y' = (a) Convert it into a system of first order ODEs like we did in class. Y = 21 1 0 2 -1 0 0 -1 * (b) Use this system to find the general solution to our vector ODE. Please, write your answer in the form * * * * 1 -() Y + C₁ C₂ C3 * + * *(a) Find the general solution of the given system of equations. x' x = (₁ (² 3 -2 10 -6 x(t) = C₁ (3) ? X (b) Assume C₁, C2 0. As t → ∞, Choose one ▼ +C₂ ? (²) ? (c) Drag the point to select the correct direction field for the given system of equations. Y X Choose one(2) Sketch the direction field associated to the system I'=x²-y-3, y = y + x² - 5. Include the x-nullclines, y-nullclines, equilibria and ordinal directions in the remaining regions.
- Find two linearly independent solutions of 2a?y" – xy' + (5x + 1)y= 0, x > 0 of the form Y1 = x" (1+ a1r + a2x? + a3x³+..) Y2 = x" (1+ bịT + bzx² + b3x³+..) where r1 > r2- Enter T1 a1 a2 a3 r2 = %3D b2 b3The direction field for = 25y is shown to the right. (a) Verify that the straight lines y = +x are solution curves, provided x + 0. (b) Sketch the solution curve with initial condition y(0) = - 1. (c) Sketch the solution curve with initial condition y(3) = 1.