Trouble Solving y'+2y=g(t): Seeking Guidance

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In summary, the conversation is about solving a differential equation with a given g(t) function and initial condition. The speaker has found a solution for the given g(t) function but is having trouble with the part where g(t) equals 0. They also mention needing direction for what properties the graph of y(t) should have at t=1.
  • #1
newtomath
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I am having trouble with part of this question:

y'+2y= g(t) where g(t) =1 for 0<t<1 and g(t)= 0 t>1 and y(0)=0

I found y= 1/2 - (1/2 * e^-2t) for 0<t<1 but am having trouble with g(t)= 0.

I know y = Ce^-2t. y(0)=0 doesn't apply here since t>1 correct? Can you point me in the right direction?
 
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  • #2
newtomath said:
I am having trouble with part of this question:

y'+2y= g(t) where g(t) =1 for 0<t<1 and g(t)= 0 t>1 and y(0)=0

I found y= 1/2 - (1/2 * e^-2t) for 0<t<1 but am having trouble with g(t)= 0.

I know y = Ce^-2t. y(0)=0 doesn't apply here since t>1 correct? Can you point me in the right direction?

What properties do you think the graph of y(t) should have at t=1?
 
  • #3
should be undefined correct?
 

FAQ: Trouble Solving y'+2y=g(t): Seeking Guidance

What is the purpose of solving the equation y'+2y=g(t)?

The purpose of solving this equation is to find a solution for the dependent variable y, which represents a changing quantity over time, given a known function g(t) and its derivative y'. This type of equation is commonly used in physics and engineering to model dynamic systems.

What is the standard approach to solving this type of equation?

The standard approach is to use an integrating factor, which is a function that helps to simplify the equation by canceling out the coefficient of the first derivative term. This allows for the use of basic integration techniques to find the solution for y.

Are there any specific conditions that need to be met in order to solve this equation?

Yes, the equation must be a first-order linear differential equation, meaning that the highest derivative term is of degree one and all other terms are linear functions (no products or powers). The function g(t) must also be continuous and the given initial condition must be specified.

Can this equation be solved analytically or does it require numerical methods?

This equation can be solved analytically using the standard approach mentioned above. However, in some cases where the function g(t) is complex or difficult to integrate, numerical methods may be used to find an approximate solution.

How can I check if my solution for y is correct?

You can check your solution by plugging it back into the original equation and verifying that it satisfies the equation for all values of t. Additionally, you can check if your solution satisfies the given initial condition. If both of these conditions are met, then your solution is likely correct.

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