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9. With the same interest rate of a = 0.1, assume that a constant withdrawal rate of $10 per unit of time is made, starting at t= 5. Find the general solution y(t) (leaving the initial deposit y(0) as a free parameter). Sketch a graph of the solution for a few different choices of y(0). Does this equation have any steady states?
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9. With the same interest rate of a = 0.1, assume that a constant with...
General Solution:
To find the general solution y(t), we can use the formula for compound interest with regular withdrawals:
y(t) = (y(0) + 10/a) * (1 + a)^t - 10/a
where y(0) is the initial deposit and a is the interest rate.
Sketching the Graph:
To sketch the graph of the solution, we can choose different values of y(0) and plot the corresponding values of y(t) for different values of t.
For example, let's choose y(0) = 50, 100, and 150.
For y(0) = 50, the graph would start at y(0) and decrease over time due to the constant withdrawal of $10 per unit of time. The graph would approach a steady state where it levels off and remains constant.
For y(0) = 100, the graph would start at a higher initial deposit and decrease at a slower rate compared to y(0) = 50. Again, it would approach a steady state.
For y(0) = 150, the graph would start even higher and decrease at an even slower rate. It would also approach a steady state.
Steady States:
This equation does have steady states. A steady state refers to a point where the value of y(t) remains constant over time.
In this case, the steady state occurs when the withdrawals equal the interest earned. The withdrawals are constant at $10 per unit of time, and the interest earned is given by the formula y(t) - (y(0) + 10/a) * (1 + a)^(t-1).
To find the steady state, we set the withdrawals equal to the interest earned:
10 = y(t) - (y(0) + 10/a) * (1 + a)^(t-1)
Simplifying this equation, we can solve for y(t):
y(t) = (y(0) + 10/a) * (1 + a)^(t-1) + 10
This equation gives us the value of y(t) at the steady state. The steady state occurs when the value of y(t) remains constant and does not change over time.
To find the general solution y(t), we can use the formula for compound interest with regular withdrawals:
y(t) = (y(0) + 10/a) * (1 + a)^t - 10/a
where y(0) is the initial deposit and a is the interest rate.
Sketching the Graph:
To sketch the graph of the solution, we can choose different values of y(0) and plot the corresponding values of y(t) for different values of t.
For example, let's choose y(0) = 50, 100, and 150.
For y(0) = 50, the graph would start at y(0) and decrease over time due to the constant withdrawal of $10 per unit of time. The graph would approach a steady state where it levels off and remains constant.
For y(0) = 100, the graph would start at a higher initial deposit and decrease at a slower rate compared to y(0) = 50. Again, it would approach a steady state.
For y(0) = 150, the graph would start even higher and decrease at an even slower rate. It would also approach a steady state.
Steady States:
This equation does have steady states. A steady state refers to a point where the value of y(t) remains constant over time.
In this case, the steady state occurs when the withdrawals equal the interest earned. The withdrawals are constant at $10 per unit of time, and the interest earned is given by the formula y(t) - (y(0) + 10/a) * (1 + a)^(t-1).
To find the steady state, we set the withdrawals equal to the interest earned:
10 = y(t) - (y(0) + 10/a) * (1 + a)^(t-1)
Simplifying this equation, we can solve for y(t):
y(t) = (y(0) + 10/a) * (1 + a)^(t-1) + 10
This equation gives us the value of y(t) at the steady state. The steady state occurs when the value of y(t) remains constant and does not change over time.
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9. With the same interest rate of a = 0.1, assume that a constant withdrawal rate of $10 per unit of time is made, starting at t= 5. Find the general solution y(t) (leaving the initial deposit y(0) as a free parameter). Sketch a graph of the solution for a few different choices of y(0). Does this equation have any steady states?
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9. With the same interest rate of a = 0.1, assume that a constant withdrawal rate of $10 per unit of time is made, starting at t= 5. Find the general solution y(t) (leaving the initial deposit y(0) as a free parameter). Sketch a graph of the solution for a few different choices of y(0). Does this equation have any steady states? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about 9. With the same interest rate of a = 0.1, assume that a constant withdrawal rate of $10 per unit of time is made, starting at t= 5. Find the general solution y(t) (leaving the initial deposit y(0) as a free parameter). Sketch a graph of the solution for a few different choices of y(0). Does this equation have any steady states? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 9. With the same interest rate of a = 0.1, assume that a constant withdrawal rate of $10 per unit of time is made, starting at t= 5. Find the general solution y(t) (leaving the initial deposit y(0) as a free parameter). Sketch a graph of the solution for a few different choices of y(0). Does this equation have any steady states?.
9. With the same interest rate of a = 0.1, assume that a constant withdrawal rate of $10 per unit of time is made, starting at t= 5. Find the general solution y(t) (leaving the initial deposit y(0) as a free parameter). Sketch a graph of the solution for a few different choices of y(0). Does this equation have any steady states? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about 9. With the same interest rate of a = 0.1, assume that a constant withdrawal rate of $10 per unit of time is made, starting at t= 5. Find the general solution y(t) (leaving the initial deposit y(0) as a free parameter). Sketch a graph of the solution for a few different choices of y(0). Does this equation have any steady states? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 9. With the same interest rate of a = 0.1, assume that a constant withdrawal rate of $10 per unit of time is made, starting at t= 5. Find the general solution y(t) (leaving the initial deposit y(0) as a free parameter). Sketch a graph of the solution for a few different choices of y(0). Does this equation have any steady states?.
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