Question

Consider the autonomous first-order differential equationand the initial condition. By hand, sketch the graph of a typical solution y(x) when y0has the given values.

(a) (b)

(c) (d)

Short answer

Answer:

The critical points are located at,and.

Step-by-step solution

Definition of differential equation.

An equation containing the derivatives of one or more unknown functions (or dependent variables), with respect to one or more independent variables, is said to be a differential equation (DE).

Critical points and phase portrait.

The given differential equation can be written as:

To find the critical points, equate,

Hence, the critical points are located at,

The phase portrait is shown below,

Graph of the solution part a).

Now we can easily sketch the solutions that satisfy (a), (b), (c), (d).

(a)., the graph is shown below,

Graph of the solution part b).

b).the graph is shown below,

Graph of the solution part c).

c)., the graph is shown below,

Graph of the solution part d).

d)., the graph is shown below,

Therefore, the nullclines to make it easier.

Therefore, the critical points are located at,and.