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3. Consider the following second order differential equation: If we let u y andv-y then express the second order differ...
Question 3 (1 mark) Attempt 1 Consider the following second order differential equation: 2 6 d2y...
Question 3 (1 mark) Attempt 1 Consider the following second order differential equation: 2 6 d2y +4 dr2 If we let u y and v y' then as an AUTONOMOUS first order system, the second order differential equation is correctly expressed as: dy d.r -y 1+112, with y(1) 9 and y'(1)5 d.r du 6 А dt1, to)1, dtv, u(to)=9, -1+11r2+2u-4v2, v(to)=5. du 6 B dry )9, ar=1+112+y-4(y), v(1)5 du d.r dt du C 1, (1) 0, =v, u(1)=9, -1+11r2+u-4v2, v(1)=5...
3. (a) Express the following ordinary differential equation and initial conditions as an autonomous system of...
3. (a) Express the following ordinary differential equation and initial conditions as an autonomous system of first order equations: 2"-223z = 2, '(0)= 1 z(0) 0, (b) Consider the following second order explicit Runge-Kutta scheme written in au- tonomous vector form (y' = f(y)): hf (ynk kihf (yn), k2 yn+1 ynk2. IT Use the second order explicit Runge-Kutta scheme with steplength h compute approximations to z(0.1) and z'(0.1) 0.1 to _
3. (a) Express the following ordinary differential equation and...
A SYSTEM MODELED WITH THE SECOND ORDER DIFFERENTIAL EQUATION PRESENTS THE FOLLOWING RESPONSE TO ZERO STATE y(t) = u(t)...
A SYSTEM MODELED WITH THE SECOND ORDER DIFFERENTIAL EQUATION PRESENTS THE FOLLOWING RESPONSE TO ZERO STATE y(t) = u(t) eatcos(wt)u(t) a,uweR WHEN ENTRANCE IS A UNITARY STAGE (u(t)). SHOWS THAT THE RESPONSE TO IMPULSE IS e-at sen(wt)u(t) a,wE R y(t)aeacos(wt)u(t) +we'
A SYSTEM MODELED WITH THE SECOND ORDER DIFFERENTIAL EQUATION PRESENTS THE FOLLOWING RESPONSE TO ZERO STATE y(t) = u(t) eatcos(wt)u(t) a,uweR WHEN ENTRANCE IS A UNITARY STAGE (u(t)). SHOWS THAT THE RESPONSE TO IMPULSE IS e-at sen(wt)u(t) a,wE R...
An autonomous system of two first order differential equations can be written as: A third order...
An autonomous system of two first order differential equations can be written as: A third order explicit Runge-Kutta scheme for an autonomous system of two first order equations is Consider the following second order differential equation, Use the Runge-Kutta scheme to find an approximate solutions of the second order differential equation, at t = 1.2, if the step size h = 0.1. Maintain at least eight decimal digit accuracy throughout all your calculations. You may express your answer as a...
Consider the nonlinear second-order differential equation x4 3(x')2 + k2x2 - 1 = 0, _ where k > 0 is a constant....
Consider the nonlinear second-order differential equation x4 3(x')2 + k2x2 - 1 = 0, _ where k > 0 is a constant. Answer to the following questions. (a) Derive a plane autonomous system from the given equation and find all the critical points (b) Classify(discriminate/categorize) all the critical points into one of the three cat- egories: stable / saddle unstable(not saddle)} (c) Show that there is no periodic solution in a simply connected region {(r, y) R2< 0} R =...
Consider the nonlinear second-order differential equation where k > 0 is a constant. Answer to the following questio...
Consider the nonlinear second-order differential equation where k > 0 is a constant. Answer to the following questions (a) Derive a plane autonomous system from the given equation and find all the critical points (b) Classify(discriminate/categorize) all the critical points into one of the three cat- egories: {stable / saddle / unstable(not saddle)) (c) Show that there is no periodic solution in a simply connected region (Hint: Use the corollary to Theorem 11.5.1)
Consider the nonlinear second-order differential equation where...
An autonomous system of two first order differential equations can be written as: A third order e...
An autonomous system of two first order differential equations can be written as: A third order explicit Runge-Kutta scheme for an autonomous system of two first order equations is hg(un,vn), 63-hf(un+2k2-k㎶n +212-11), 13 hg(un+2k2-ki,un +212-4), t-4 Consider the following second order differential equation, +2dy-7y2-12, with y(0)= 4 and y'(0)=0. dt2 dt Use the Runge-Kutta scheme to find an approximate solution of the second order differential equation, at t = 0.1, if the step size h = 0.05 Maintain at least...
A system of two first order differential equations can be written as: A second order explicit R...
A system of two first order differential equations can be written as: A second order explicit Runge-Kutta scheme for the system of two first order equations is Consider the following second order differential equation: Use the Runge-kutta scheme to find an approximate solution of the second order differential equation, at x = 0.2, if the step size h = 0.1. Maintain at least eight decimal digit accuracy throughout all your calculations. You may express your answer as a five decimal...
[Reduction of Order] Explain how a second order differential equation t^(2)y′′ + bty′ + ct^(3)y =...
[Reduction of Order] Explain how a second order differential equation t^(2)y′′ + bty′ + ct^(3)y = f can be converted into a first order system, and the Euler method for that system.
Find a first-order system of ordinary differential equations equivalent to the second-order nonlinear ordinary differential equation...
Find a first-order system of ordinary differential equations
equivalent to the second-order nonlinear ordinary differential
equation y ^-- = 3y 0 + (y 3 − y)
(3 points) Find a first-order system of ordinary differential equations equivalent to the second-order nonlinear ordinary differential equation y" = 3y' +(y3 – y).
Question 3 (1 mark) Attempt 1 Consider the following second order differential equation: 2 6 d2y +4 dr2 If we let u y and v y' then as an AUTONOMOUS first order system, the second order differential equation is correctly expressed as: dy d.r -y 1+112, with y(1) 9 and y'(1)5 d.r du 6 А dt1, to)1, dtv, u(to)=9, -1+11r2+2u-4v2, v(to)=5. du 6 B dry )9, ar=1+112+y-4(y), v(1)5 du d.r dt du C 1, (1) 0, =v, u(1)=9, -1+11r2+u-4v2, v(1)=5...
3. (a) Express the following ordinary differential equation and initial conditions as an autonomous system of first order equations: 2"-223z = 2, '(0)= 1 z(0) 0, (b) Consider the following second order explicit Runge-Kutta scheme written in au- tonomous vector form (y' = f(y)): hf (ynk kihf (yn), k2 yn+1 ynk2. IT Use the second order explicit Runge-Kutta scheme with steplength h compute approximations to z(0.1) and z'(0.1) 0.1 to _
3. (a) Express the following ordinary differential equation and...
A SYSTEM MODELED WITH THE SECOND ORDER DIFFERENTIAL EQUATION PRESENTS THE FOLLOWING RESPONSE TO ZERO STATE y(t) = u(t) eatcos(wt)u(t) a,uweR WHEN ENTRANCE IS A UNITARY STAGE (u(t)). SHOWS THAT THE RESPONSE TO IMPULSE IS e-at sen(wt)u(t) a,wE R y(t)aeacos(wt)u(t) +we'
A SYSTEM MODELED WITH THE SECOND ORDER DIFFERENTIAL EQUATION PRESENTS THE FOLLOWING RESPONSE TO ZERO STATE y(t) = u(t) eatcos(wt)u(t) a,uweR WHEN ENTRANCE IS A UNITARY STAGE (u(t)). SHOWS THAT THE RESPONSE TO IMPULSE IS e-at sen(wt)u(t) a,wE R...
Consider the nonlinear second-order differential equation x4 3(x')2 + k2x2 - 1 = 0, _ where k > 0 is a constant. Answer to the following questions. (a) Derive a plane autonomous system from the given equation and find all the critical points (b) Classify(discriminate/categorize) all the critical points into one of the three cat- egories: stable / saddle unstable(not saddle)} (c) Show that there is no periodic solution in a simply connected region {(r, y) R2< 0} R =...
Consider the nonlinear second-order differential equation where k > 0 is a constant. Answer to the following questions (a) Derive a plane autonomous system from the given equation and find all the critical points (b) Classify(discriminate/categorize) all the critical points into one of the three cat- egories: {stable / saddle / unstable(not saddle)) (c) Show that there is no periodic solution in a simply connected region (Hint: Use the corollary to Theorem 11.5.1)
Consider the nonlinear second-order differential equation where...
An autonomous system of two first order differential equations can be written as: A third order explicit Runge-Kutta scheme for an autonomous system of two first order equations is hg(un,vn), 63-hf(un+2k2-k㎶n +212-11), 13 hg(un+2k2-ki,un +212-4), t-4 Consider the following second order differential equation, +2dy-7y2-12, with y(0)= 4 and y'(0)=0. dt2 dt Use the Runge-Kutta scheme to find an approximate solution of the second order differential equation, at t = 0.1, if the step size h = 0.05 Maintain at least...
Find a first-order system of ordinary differential equations
equivalent to the second-order nonlinear ordinary differential
equation y ^-- = 3y 0 + (y 3 − y)
(3 points) Find a first-order system of ordinary differential equations equivalent to the second-order nonlinear ordinary differential equation y" = 3y' +(y3 – y).
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