# How Does the Initial Value y0 Affect the Solution Interval of dy/dt + y^3 = 0?

• jwqwerty
In summary, the interval in which the solution exists depends on the initial value y0, with y=0 for all values of t when y0=0, and y having two solutions, y=±y0/√(2ty0^(2)+1), in the interval ∞>t>-1/2y0^2 when y0≠0.
jwqwerty

## Homework Statement

solve dy/dt+y^3=0 , y(0)=y0 and determine how the interval in which the solution exists depends on the initial value y0

## Homework Equations

Theorem: Let the fuctions f and df/dy(partial differentiation) be continuous in some rectangle a<t<b , c<y<d containing (t0,y0). Then in some interval t0-h<t<t0+h contained in a<t<b, there is a unique solution y=∅(t) of the initial value problem dy/dt=f(t,y), y(t0)=y0

## The Attempt at a Solution

According to the theorem above, by solving dy/dt+y^3=0 y should only have one solution (y) in some interval.

dy/dt+y^3=0

i) when y≠0
-y^(-3)dy=dt and integrate both sides
0.5y^(-2)=t+c (c is a constant)

by initial condition y(0)=y0
0.5y^(-2)=t+0.5y0^(-2) and multiply both sides by 2
y^(-2)=2t+y0^(-2)
y^2=1/(2t+y0^(-2))
=y0^2/(2ty0^(2)+1)
Thus, y= ±y0/√(2ty0^(2)+1) (∞>t>-1/2y0^2 since numbers inside root must be bigger than 0)

ii) y=0
y=0 holds true for all t.

1. y has three solutions when ∞>t>-1/2y0^2 which are y= ±y0/√(2ty0^(2)+1), y=0.

did i do something wrong?

Last edited:

No, your solution is correct. The initial value y0 affects the solution in two ways:
1. If y0=0, then y=0 for all values of t.
2. If y0≠0, then y has two solutions, y=±y0/√(2ty0^(2)+1), in the interval ∞>t>-1/2y0^2.

## 1. What is a differential question?

A differential question is a type of question that asks for a comparison or contrast between two or more things. It often involves looking at the differences or changes between the variables being studied.

## 2. How is a differential question different from other types of questions?

Differential questions focus on the differences or changes between variables, while other types of questions may focus on the similarities, causes, or effects of those variables.

## 3. What is the purpose of asking a differential question?

The purpose of a differential question is to gain a deeper understanding of the relationship between variables and to identify any patterns or trends that may exist.

## 4. Can a differential question be answered through experimentation?

Yes, many differential questions can be answered through experimentation. By manipulating the variables and observing the changes, scientists can gather data to help answer the question.

## 5. How do differential questions contribute to scientific research?

Differential questions are important in scientific research as they allow scientists to analyze and compare different variables, leading to a better understanding of how they are related. This can help in making predictions, developing theories, and furthering scientific knowledge.

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