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yifli
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Given a differential equation [itex]dy/dt = F(t,y)[/itex], can anyone give me an example that shows no solution exists if [itex]F(t,y)[/itex] is discontinuous?
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Does a Solution Exist for a Discontinuous Differential Equation?
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In summary, a scientific experiment is a test conducted to gather data and support or refute a hypothesis, such as testing the effects of different fertilizers on plant growth. A scientific theory is a well-supported explanation for a natural phenomenon, like the theory of evolution. A scientific law is a rule that describes a natural pattern or behavior, such as Newton's Law of Universal Gravitation. A scientific hypothesis is a testable prediction based on observations, such as the effect of soil type on plant growth. A scientific model is a representation of a complex system, such as a climate model used to predict future climate patterns.
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Hi yifli! 
Check out Darboux's theorem. It says that the derivative of every function have the intermediate value property.
So, if you can find a function f without that property, then you'd know that
[tex]y^\prime=f(t)[/tex]
has no solutions. So, try to find a function without that property...
Check out Darboux's theorem. It says that the derivative of every function have the intermediate value property.
So, if you can find a function f without that property, then you'd know that
[tex]y^\prime=f(t)[/tex]
has no solutions. So, try to find a function without that property...
1. What is an example of a scientific experiment?
An example of a scientific experiment could be testing the effects of different fertilizer types on plant growth. This experiment would involve setting up different groups of plants, using different types of fertilizer on each group, and measuring the growth of the plants over a set period of time.
2. Can you give me an example of a scientific theory?
One example of a scientific theory is the theory of evolution, which explains how species change over time through the process of natural selection. This theory is supported by a large body of evidence from various scientific fields, including genetics, paleontology, and ecology.
3. What is an example of a scientific law?
A classic example of a scientific law is Newton's Law of Universal Gravitation, which states that all objects in the universe are attracted to each other by a force called gravity. This law has been tested and confirmed countless times and is an essential concept in physics.
4. Can you provide an example of a scientific hypothesis?
An example of a scientific hypothesis could be "If plant growth is affected by the type of soil used, then plants grown in nutrient-rich soil will grow taller than those grown in nutrient-poor soil." This hypothesis can then be tested through a controlled experiment to determine if it is supported or not.
5. What is an example of a scientific model?
A common example of a scientific model is a climate model, which uses computer simulations to predict future climate patterns based on various factors such as greenhouse gas emissions and solar radiation. These models help scientists understand and make predictions about complex systems like the Earth's climate.
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