Homework Answers
1.
2
We know that a differential equation or system of ordinary
differential equations is said to be autonomous if it does not
explicitly contain the independent variable (usually denoted
).
Here clearly differential equation must be like of the form
And these contain t as independent variables here.
So we can say the ODE system cannot be autonomous.
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Suppose that a first-order, two-variable ODE system has unique solutions for any initial value and one...
2. (8 points) Solve the linear, 1st order ODE with initial value: dy dr 3. (7...
2. (8 points) Solve the linear, 1st order ODE with initial value: dy dr 3. (7 points) Find all critical points and the phase portrait of the autonomous Ist order ODE dy dr -5y+4 Classify each critical point as asymptotically stable, unstable or semi-stable. Sketch typical solution curves in the regions in the ry plane separated by equilibrium solutions. dy dx (S points) Solve the Bernoulli equation:-(- 31-1 7. (8 points) Solve the ODE by variation of parameters: -4y+4y (+...
Le-t are solutions of a second-order /2e5t and y2(t) Suppose y1(t) = homogeneous linear ODE on R. Which one of the foll...
Le-t are solutions of a second-order /2e5t and y2(t) Suppose y1(t) = homogeneous linear ODE on R. Which one of the following is also a solution to the same ODE? y(t) e5t-2 y(t) ee y(t) e5t e 1 y(t) 2e5t
Le-t are solutions of a second-order /2e5t and y2(t) Suppose y1(t) = homogeneous linear ODE on R. Which one of the following is also a solution to the same ODE? y(t) e5t-2 y(t) ee y(t) e5t e 1 y(t) 2e5t
Suppose that a simple spring-mass system can be modeled by the 2nd-order Non- homogeneous ODE stated...
Suppose that a simple spring-mass system can be modeled by the 2nd-order Non- homogeneous ODE stated below. Answer the following questions concerning the properties of this spring-mass system. 4 + 4 = 2 cos(2t); }(0) = 4 (0) = 0 (i) is the spring-mass system an underdamped system, critcally-damped system, overdamped system, or a system with no damping? [Select] (ii) Can this system ever achieve resonance? Select] (iii) is the spring-mass system characterized by the ODE stable? Select] (iv) Does...
Read the sample Matlab code euler.m. Use either this code, or write your own code, to solve first...
Read the sample Matlab code euler.m. Use either this code, or write your own code, to solve first order ODE = f(t,y) dt (a). Consider the autonomous system Use Euler's method to solve the above equation. Try different initial values, plot the graphs, describe the behavior of the solutions, and explain why. You need to find the equilibrium solutions and classify them. (b). Numerically solve the non-autonomous system dy = cost Try different initial values, plot the graphs, describe the...
ODE - 1 Q.1 Solve the following first order linear initial value problems. (a) 2ndp -...
ODE
- 1
Q.1 Solve the following first order linear initial value problems. (a) 2ndp - 0.4pdt -0, p(1)- 0.2 (b) v(f) dv (1) +*dt - 0, v(2) -2 + 2v ()- 6, v(0) - 10 (c) (d) The first order differential equation, initial value problem, - Sms, v(0) = 2ms. describes the motion of a car. Find an expression for the speed v () and determine the velocity of the car after 10 seconds from the beginning of its...
Consider the plane autonomous system 4) 2 X'=AX with A (a) Find two linearly independent real solutions of the syst...
Consider the plane autonomous system 4) 2 X'=AX with A (a) Find two linearly independent real solutions of the system (b) Classify the stability (stable or unstable) and the type (center, node, saddle, or spiral) of the critical point (0,0). (c) Plot the phase portrait of the system containing a trajectory with direction as t-oo whose initial value is X(0) (0,6)7 and any other trajectory with direc- tion. (You do not need to draw solution curves explicitly.)
Consider the plane...
1. Solve the initial value problem for a damped mass-spring system acted upon by a sinusoidal...
1. Solve the initial value problem for a damped mass-spring system acted upon by a sinusoidal force for some time interval f(t) = {10 sin 2t 0 0<t< y(0) 1, y'(0) -5 y"2y' 2y f(t), Tt zusor= 2. Consider two masses and three springs without no external force. The resulting force balance can be expressed as two second order ODES shown as below. mx=-(k k2)x1+ kzx2 m2x2 (k2k3)x2 + k2x1 15 If m 2,m2 ki = 1,k2 = 3, k3...
7. Answer the questions below for the following initial value problem: y (t) = sin y,...
7. Answer the questions below for the following initial value problem: y (t) = sin y, 0 <y(0) < 27. (a) [1 pt) Determine the equilibrium (i.e., critical or steady-state) solutions. (b) (2 pts) Construct a sign chart for y' = sin y. Hy' = sin y 21 (c) (3 pts] Now construct a sign chart for y", and find the inflection points (if any). Hy" = f(y) 271 (d) [5 pts] Draw the phase line, and sketch a graph...
please show all your work . (6 points) Of the four initial or boundary value problems below, ouly one is guaranteed...
please show all your work
. (6 points) Of the four initial or boundary value problems below, ouly one is guaranteed to have a unique solution according to the Existence and Uniqueness Theorons. Which one i i (a) ty"-Py, + e'y = ), y(1)s 0, V(1) = T. tan (f (b) ty" + 2/-3y = 0, y (0)0. y(0) = 2, y(5) = 0. (d) V, + sec(t)y = sin(2t),
. (6 points) Of the four initial or boundary value...
Exercise 5.27 Suppose and 2t) are solutions of a linear homogeneous system A (t)x with a coeffici...
Exercise 5.27 Suppose and 2t) are solutions of a linear homogeneous system A (t)x with a coefficient matrix A(t) that is continuous on an interval a < t < β. Prove that the determinant s() -det( 3) (t) 2 is either never equal to 0 for α < t < β or else it is identically (i.e., alu ays) equal to 0 on α < t < β. (Hint: by direct calculation show the determinant satisfies the first order, linear...
2. (8 points) Solve the linear, 1st order ODE with initial value: dy dr 3. (7 points) Find all critical points and the phase portrait of the autonomous Ist order ODE dy dr -5y+4 Classify each critical point as asymptotically stable, unstable or semi-stable. Sketch typical solution curves in the regions in the ry plane separated by equilibrium solutions. dy dx (S points) Solve the Bernoulli equation:-(- 31-1 7. (8 points) Solve the ODE by variation of parameters: -4y+4y (+...
Le-t are solutions of a second-order /2e5t and y2(t) Suppose y1(t) = homogeneous linear ODE on R. Which one of the following is also a solution to the same ODE? y(t) e5t-2 y(t) ee y(t) e5t e 1 y(t) 2e5t
Le-t are solutions of a second-order /2e5t and y2(t) Suppose y1(t) = homogeneous linear ODE on R. Which one of the following is also a solution to the same ODE? y(t) e5t-2 y(t) ee y(t) e5t e 1 y(t) 2e5t
Suppose that a simple spring-mass system can be modeled by the 2nd-order Non- homogeneous ODE stated below. Answer the following questions concerning the properties of this spring-mass system. 4 + 4 = 2 cos(2t); }(0) = 4 (0) = 0 (i) is the spring-mass system an underdamped system, critcally-damped system, overdamped system, or a system with no damping? [Select] (ii) Can this system ever achieve resonance? Select] (iii) is the spring-mass system characterized by the ODE stable? Select] (iv) Does...
Read the sample Matlab code euler.m. Use either this code, or write your own code, to solve first order ODE = f(t,y) dt (a). Consider the autonomous system Use Euler's method to solve the above equation. Try different initial values, plot the graphs, describe the behavior of the solutions, and explain why. You need to find the equilibrium solutions and classify them. (b). Numerically solve the non-autonomous system dy = cost Try different initial values, plot the graphs, describe the...
ODE
- 1
Q.1 Solve the following first order linear initial value problems. (a) 2ndp - 0.4pdt -0, p(1)- 0.2 (b) v(f) dv (1) +*dt - 0, v(2) -2 + 2v ()- 6, v(0) - 10 (c) (d) The first order differential equation, initial value problem, - Sms, v(0) = 2ms. describes the motion of a car. Find an expression for the speed v () and determine the velocity of the car after 10 seconds from the beginning of its...
Consider the plane autonomous system 4) 2 X'=AX with A (a) Find two linearly independent real solutions of the system (b) Classify the stability (stable or unstable) and the type (center, node, saddle, or spiral) of the critical point (0,0). (c) Plot the phase portrait of the system containing a trajectory with direction as t-oo whose initial value is X(0) (0,6)7 and any other trajectory with direc- tion. (You do not need to draw solution curves explicitly.)
Consider the plane...
1. Solve the initial value problem for a damped mass-spring system acted upon by a sinusoidal force for some time interval f(t) = {10 sin 2t 0 0<t< y(0) 1, y'(0) -5 y"2y' 2y f(t), Tt zusor= 2. Consider two masses and three springs without no external force. The resulting force balance can be expressed as two second order ODES shown as below. mx=-(k k2)x1+ kzx2 m2x2 (k2k3)x2 + k2x1 15 If m 2,m2 ki = 1,k2 = 3, k3...
7. Answer the questions below for the following initial value problem: y (t) = sin y, 0 <y(0) < 27. (a) [1 pt) Determine the equilibrium (i.e., critical or steady-state) solutions. (b) (2 pts) Construct a sign chart for y' = sin y. Hy' = sin y 21 (c) (3 pts] Now construct a sign chart for y", and find the inflection points (if any). Hy" = f(y) 271 (d) [5 pts] Draw the phase line, and sketch a graph...
please show all your work
. (6 points) Of the four initial or boundary value problems below, ouly one is guaranteed to have a unique solution according to the Existence and Uniqueness Theorons. Which one i i (a) ty"-Py, + e'y = ), y(1)s 0, V(1) = T. tan (f (b) ty" + 2/-3y = 0, y (0)0. y(0) = 2, y(5) = 0. (d) V, + sec(t)y = sin(2t),
. (6 points) Of the four initial or boundary value...
Exercise 5.27 Suppose and 2t) are solutions of a linear homogeneous system A (t)x with a coefficient matrix A(t) that is continuous on an interval a < t < β. Prove that the determinant s() -det( 3) (t) 2 is either never equal to 0 for α < t < β or else it is identically (i.e., alu ays) equal to 0 on α < t < β. (Hint: by direct calculation show the determinant satisfies the first order, linear...
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