# Differential equations, qualitative solution

• MaxManus
In summary, the text discusses a solution to dy/dt = -ty^2 with initial condition y(0) = 1. It is noted that the solution is decreasing and never reaches zero, as y(t) = 0 is an equilibrium solution. The derivative of the solution is also examined and it is determined that y(t) can never be zero for all values of t. This is supported by the fact that setting y(t) = 0 would result in a false equality.

#### MaxManus

dy/dt = -ty^2 y(0) = 1
What can you say about the solution over the interval 0<=t <=2?

The text says that the solution is decreasing and never zero because y(t) = 0 for all t is an equilibrium solution.

I can see why the solution can never cross the x-axis, but why can't the solution become zero?

What did you get for the solution y(t)?

I got y = $$\frac{1}{t^2 + 1}$$, but I got the impression that looking at the derivative was enough to make the conclusion

Anyone?

MaxManus said:
Anyone?

if it crosses the t-axis, then shouldn't y=0 ? and what does that mean?

rock.freak667 said:
if it crosses the t-axis, then shouldn't y=0 ? and what does that mean?

Yes, but why can't it become zero and stay there?

MaxManus said:
Yes, but why can't it become zero and stay there?

How exactly would it become zero? That would mean that at some value of 't' the corresponding value of 'y' would be zero.

$$0 = \frac{1}{t^2+1}$$

You would end up with a false equality, which can only mean that y(t) is never zero for all values of t.

## 1. What are differential equations?

Differential equations are mathematical equations that involve one or more derivatives of an unknown function. They are used to model a variety of natural phenomena, such as growth, decay, and motion.

## 2. How are differential equations solved?

Differential equations can be solved analytically or numerically. Analytical solutions involve finding an expression for the unknown function, while numerical solutions use algorithms to approximate the solution.

## 3. What is a qualitative solution?

A qualitative solution to a differential equation involves analyzing the behavior of the solution without finding an exact numerical solution. This can include identifying equilibrium points, stability, and other characteristics of the solution.

## 4. How are differential equations used in real life?

Differential equations are used in many fields, including physics, biology, economics, and engineering. They can be used to model and predict the behavior of complex systems, such as population growth, weather patterns, and electrical circuits.

## 5. What is the significance of finding a solution to a differential equation?

Finding a solution to a differential equation allows us to understand and make predictions about the behavior of a system. It also allows us to optimize and control these systems, leading to advancements in technology and science.