Question

Write the given nonlinear second-order differential equation as a plane autonomous system.

x'' + 6 (x/(1+ x^2)) + 5x' = 0

x' = y
y' = ___________
Find all critical points of the resulting system. (x, y) = ________________

Answer 1

To write the given nonlinear second-order differential equation as a plane autonomous system, introduce a new variable y = x'. To find the critical points of the resulting system, solve the system of equations x' = 0 and y' = 0.

Step-by-step explanation:

To write the given nonlinear second-order differential equation as a plane autonomous system, we can introduce a new variable, say y, such that y = x'. Then x'' = y'. We can rewrite the given equation as:

y' + 6*(x/(1 + x²))*y + 5*x = 0

This forms a system of two first-order differential equations:

x' = y

y' = -6*(x/(1 + x²))*y - 5*x

To find the critical points of the resulting system, we need to solve the system of equations:

x' = 0

y' = 0

The critical points (x, y) are the solutions to this system.

Answer 2

y' = -6x/(1+x^2) - 5y

Critical points: (0,0), (-1,0), (1,0)

Step-by-step explanation:

Define state variables: We introduce a new variable y to represent x'. Therefore, x' = y.

Substitute y into the equation: Replace x' with y in the original equation: x'' + 6(x/(1+x^2)) + 5y = 0

Express x'' as a function of x and y: Differentiate the equation for y (x' = y) with respect to time: y' = x''. Substitute y' with the new equation for y': x'' = -6x/(1+x^2) - 5y

Plane autonomous system: Now we have the system: x' = y y' = -6x/(1+x^2) - 5y

Critical points: Critical points are where both x' and y' are zero. Solve the system for x and y: y = 0 (from x') -6x/(1+x^2) - 5y = 0 (from y') Solve for x in the second equation: x = 0, -1, 1 Substitute x back into x' = 0 to confirm y = 0 for all three values of x. Therefore, the critical points are: (0,0), (-1,0), (1,0)