How Do You Determine Values of p and q in Differential Equation Superposition?

In summary, y1[t] and y2[t] are two functions that solve different equations with different initial conditions. To find a solution to the given equation, we need to find the values of p and q that will make y[t] equal to a combination of y1[t] and y2[t]. This new equation can then be solved to find the solution to the given equation.
  • #1
Gspace
18
0

Homework Statement



y1[t] solves
y't]+9.7 y[t]=E^(-0.4 t) Cos[t], with y[0]=0,
and y2[t] solves

y'[t]+9.7 y[t]=0, with y[0]=1.

What numbers p and q do you pick to make
y[t]=p y1[t] + q y2[t]
solve
y'[t]+9.7 y[t]=3 E^(-0.4 t) Cos[t], with y[0]=3?

Homework Equations




y1[t] solves
y't]+9.7 y[t]=E^(-0.4 t) Cos[t], with y[0]=0,
and y2[t] solves

y'[t]+9.7 y[t]=0, with y[0]=1.
 
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  • #2
So we got [tex]y(t)=p y_1(t)+qy_2(t)[/tex].

What you need to do now is to plug this y(t) in you equation [tex]y^\prime(t)+9,7y(t)=3e^{- 0,4 t}[/tex]. You will obtain a nice equation this way which you need to solve...
 
  • #3
Makes sense. Thanks micromass
 

FAQ: How Do You Determine Values of p and q in Differential Equation Superposition?

What is superposition?

Superposition is a principle in physics that states that when two or more waves or quantum states are combined, the resulting wave or state is the sum of the individual waves or states. This means that multiple waves or states can exist in the same space at the same time.

How does superposition apply to quantum mechanics?

In quantum mechanics, superposition is a fundamental principle that describes the state of a quantum system. It states that a quantum system can exist in multiple states simultaneously, until it is observed or measured. This is known as the principle of superposition of states.

What is an example of superposition in everyday life?

One example of superposition in everyday life is the interference pattern created by two overlapping waves, such as when you throw two pebbles into a pond and the ripples intersect. The resulting pattern is a combination of the individual waves, showing the principle of superposition in action.

How does superposition relate to the Schrodinger's cat thought experiment?

The famous Schrodinger's cat thought experiment illustrates superposition in quantum mechanics. In this experiment, a cat is placed in a sealed box with a radioactive substance that has a 50% chance of decaying and releasing poison that would kill the cat. According to the principle of superposition, until the box is opened and the cat is observed, it exists in a superposition of both alive and dead states.

What are the implications of superposition?

Superposition has important implications for quantum computing and quantum information processing, as it allows for the storage and manipulation of information in multiple states simultaneously. It also challenges our understanding of reality and the role of observation in determining the state of a system.

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