Question

consider the autonomous first-order differential equation dy/dx = y − y3 and the initial condition y(0) = y0. sketch the graph of a typical solution y(x) when y0 has the given values.

Answer 1

To sketch the graph of the solution to the given differential equation, consider three cases: y0 > 1, 0 < y0 < 1, and y0 = 0 or y0 < 0. The graph will either converge to a stable equilibrium point, diverge to positive infinity, or diverge to negative infinity depending on the initial condition.

Step-by-step explanation:

To sketch the graph of the solution to the autonomous first-order differential equation dy/dx = y - y^3 with the initial condition y(0) = y0, we need to determine the behavior of the function y(x) for different values of y0. Let's consider three cases:

1. 
   
3. If y0 > 1, the function y(x) will converge to a stable equilibrium point at y = 1 as x approaches infinity. The graph will resemble an S-shape called a stable node.
4. 
   
6. If 0 < y0 < 1, the function y(x) will diverge to positive infinity as x approaches infinity. The graph will resemble a curve bending upward.
7. 
   
9. If y0 = 0 or y0 < 0, the function y(x) will diverge to negative infinity as x approaches infinity. The graph will resemble a curve bending downward.

Sketching the graph based on these cases will help visualize the behavior of the solution for different initial conditions.