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An unforced, undamped Duffing equation (see web and Kreyszig §4.6 for applications) can be modelled with the following system of first-order ordinary differential equations dy =y' = x(1 - Bx²) dt da = x = y₁ dt where x(t) is the displacement at time t of an oscillator and 3> 0 is a given constant. 1. What physical quantity does y(t) represent? 2. Determine the equations for the nullclines of the system. 3. Find the equilibrium points of the system. 4. Use the chain rule to derive a first-order ODE for the trajectories in the ry phase plane. Express this ODE in the form dy f(x) dx g(y)' where f(x) and g(y) are functions which you need to determine. 5. Re-write the above ODE in differential form. 6. Use the solution method for exact equations to find the general solution to the ODE in 4. You MUST use the exact method, including the test for exactness, even though the equation is separable. 7. Determine the particular solution for the phase trajectory that satisfies the initial condition x(0) = 0 and y(0) = 1.

An unforced, undamped Duffing equation (see web and Kreyszig §4.6 for applications) can be modelled with the following system of first-order ordinary differential equations dy =y' = x(1 - Bx²) dt da = x = y₁ dt where x(t) is the displacement at time t of an oscillator and 3> 0 is a given constant. 1. What physical quantity does y(t) represent? 2. Determine the equations for the nullclines of the system. 3. Find the equilibrium points of the system. 4. Use the chain rule to derive a first-order ODE for the trajectories in the ry phase plane. Express this ODE in the form dy f(x) dx g(y)' where f(x) and g(y) are functions which you need to determine. 5. Re-write the above ODE in differential form. 6. Use the solution method for exact equations to find the general solution to the ODE in 4. You MUST use the exact method, including the test for exactness, even though the equation is separable. 7. Determine the particular solution for the phase trajectory that satisfies the initial condition x(0) = 0 and y(0) = 1.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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An unforced, undamped Duffing equation (see web and Kreyszig §4.6 for applications) can be modelled with the following system of first-order ordinary differential equations dy = y' = x(1 – Bx²) dt dr L' = y, dt where r(t) is the displacement at time t of an oscillator and 3 > 0 is a given constant. 1. What physical quantity does y(t) represent? 2. Determine the equations for the nullclines of the system. 3. Find the equilibrium points of the system. 4. Use the chain rule to derive a first-order ODE for the trajectories in the xy phase plane. Express this ODE in the form dy _ f(x) g(y) ' dr where f(x) and g(y) are functions which you need to determine. 5. Re-write the above ODE in differential form. 6. Use the solution method for exact equations to find the general solution to the ODE in 4. You MUST use the exact method, including the test for exactness, even though the equation is separable. 7. Determine the particular solution for the phase trajectory that satisfies the initial condition x(0) = 0 and y(0) = 1.
Transcribed Image Text:An unforced, undamped Duffing equation (see web and Kreyszig §4.6 for applications) can be modelled with the following system of first-order ordinary differential equations dy = y' = x(1 – Bx²) dt dr L' = y, dt where r(t) is the displacement at time t of an oscillator and 3 > 0 is a given constant. 1. What physical quantity does y(t) represent? 2. Determine the equations for the nullclines of the system. 3. Find the equilibrium points of the system. 4. Use the chain rule to derive a first-order ODE for the trajectories in the xy phase plane. Express this ODE in the form dy _ f(x) g(y) ' dr where f(x) and g(y) are functions which you need to determine. 5. Re-write the above ODE in differential form. 6. Use the solution method for exact equations to find the general solution to the ODE in 4. You MUST use the exact method, including the test for exactness, even though the equation is separable. 7. Determine the particular solution for the phase trajectory that satisfies the initial condition x(0) = 0 and y(0) = 1.
An unforced, undamped Duffing equation (see web and Kreyszig §4.6 for applications) can be modelled with the following system of first-order ordinary differential equations dy = y' = x(1 – Bx²) dt dr L' = y, dt where r(t) is the displacement at time t of an oscillator and 3 > 0 is a given constant. 1. What physical quantity does y(t) represent? 2. Determine the equations for the nullclines of the system. 3. Find the equilibrium points of the system. 4. Use the chain rule to derive a first-order ODE for the trajectories in the xy phase plane. Express this ODE in the form dy _ f(x) g(y) ' dr where f(x) and g(y) are functions which you need to determine. 5. Re-write the above ODE in differential form. 6. Use the solution method for exact equations to find the general solution to the ODE in 4. You MUST use the exact method, including the test for exactness, even though the equation is separable. 7. Determine the particular solution for the phase trajectory that satisfies the initial condition x(0) = 0 and y(0) = 1.
Transcribed Image Text:An unforced, undamped Duffing equation (see web and Kreyszig §4.6 for applications) can be modelled with the following system of first-order ordinary differential equations dy = y' = x(1 – Bx²) dt dr L' = y, dt where r(t) is the displacement at time t of an oscillator and 3 > 0 is a given constant. 1. What physical quantity does y(t) represent? 2. Determine the equations for the nullclines of the system. 3. Find the equilibrium points of the system. 4. Use the chain rule to derive a first-order ODE for the trajectories in the xy phase plane. Express this ODE in the form dy _ f(x) g(y) ' dr where f(x) and g(y) are functions which you need to determine. 5. Re-write the above ODE in differential form. 6. Use the solution method for exact equations to find the general solution to the ODE in 4. You MUST use the exact method, including the test for exactness, even though the equation is separable. 7. Determine the particular solution for the phase trajectory that satisfies the initial condition x(0) = 0 and y(0) = 1.
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