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Chapter 10: Problem 8

Is the constant function \(f(t)=-4\) a solution of the differential equation \(y^{\prime}=t^{2}(y+4) ?\)

Short Answer

Expert verified
Yes, f(t) = -4 is a solution.

Step by step solution

01

Identify the Given Function

The given function is f(t) = -4.
02

Calculate the Derivative

The derivative of a constant function is 0. Therefore, the derivative of f(t) = -4 is f'(t) = 0.
03

Substitute into the Differential Equation

Substitute f(t) = -4 and f'(t) = 0 into the differential equation y' = t^2 (y + 4). We simplify to get: 0 = t^2 (-4 + 4).
04

Simplify the Equation

Simplify the right side: t^2 (-4 + 4) = t^2 (0) = 0.
05

Verify the Solution

Since both sides of the equation equal 0, the given function f(t) = -4 satisfies the differential equation y' = t^2 (y + 4).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constant Functions
Before diving into the solution, it's important to understand what a constant function is. A constant function is a function that always returns a fixed value, regardless of the input. In mathematical terms, if we have a function \(f(t)\), and it is constant, this means: \ \(f(t) = C\) for any \(t\), where \(C\) is a constant number.
In our exercise, the constant function provided is \(f(t) = -4\). It does not change no matter what value of \(t\) we input. This property makes them straightforward and predictable.
Derivatives
One of the key steps in verifying if \(f(t) = -4\) is a solution to the given differential equation involves calculating its derivative. The derivative of a function measures how it changes with respect to its input. For constant functions, this becomes very simple.
When you take the derivative of any constant function \(f(t) = C\) where \(C\) is a constant number, the result is always zero: \ \(f'(t) = \frac{d}{dt}C = 0\).
This is because constants don鈥檛 change, thus their rate of change is zero. In our case, the derivative of \(f(t) = -4\) is \(\frac{d}{dt}(-4) = 0\). This zero derivative becomes crucial when substituting into the differential equation later.
Verification of Solutions
After calculating the derivative of the function, the next step is verification. This ensures that the constant function \(f(t) = -4\) fulfills the requirements of the differential equation \(y' = t^2(y + 4)\).
Substituting \(f(t) = -4\) and \(f'(t) = 0\) into the equation, we get: \ \(0 = t^2(-4 + 4)\).
Simplifying within the parentheses first, we have: \ \(-4 + 4 = 0\).
So the equation becomes: \ \(0 = t^2(0) = 0\).
Since both sides of the equation equal zero, the function \(f(t) = -4\) is indeed a valid solution to the differential equation.
Verification is a crucial step in differential equations as it confirms the correctness of a proposed solution.

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Most popular questions from this chapter

A company wishes to set aside funds for future expansion and so arranges to make continuous deposits into a savings account at the rate of \(\$ 10,000\) per year. The savings account earns \(5 \%\) interest compounded continuously. (a) Set up the differential equation that is satisfied by the amount \(f(t)\) of money in the account at time \(t.\) (b) Solve the differential equation in part (a), assuming that \(f(0)=0,\) and determine how much money will be in the account at the end of 5 years.

The Los Angeles Zoo plans to transport a California sea lion to the San Diego Zoo. The animal will be wrapped in a wet blanket during the trip. At any time t, the blanket will lose water (due to evaporation) at a rate proportional to the amount \(f(t)\) of water in the blanket, with constant of proportionality \(k=-.3 .\) Initially, the blanket will contain 2 gallons of seawater. (a) Set up the differential equation satisfied by \(f(t).\) (b) Use Euler's method with \(n=2\) to estimate the amount of moisture in the blanket after 1 hour. (c) Solve the differential equation in part (a) and compute \(f(1).\) (d) Compare the answers in parts (b) and (c) and approximate the error in using Euler's method.

A person planning for her retirement arranges to make continuous deposits into a savings account at the rate of \(\$ 3600\) per year. The savings account earns \(5 \%\) interest compounded continuously. (a) Set up a differential equation that is satisfied by \(f(t),\) the amount of money in the account at time \(t.\) (b) Solve the differential equation in part (a), assuming that \(f(0)=0,\) and determine how much money will be in the account at the end of 25 years.

Solve the following differential equations: $$y^{\prime}=\frac{\ln t}{t y}$$

Amount of Information Learned In certain learning situations a maximum amount, \(M,\) of information can be learned, and at any time, the rate of learning is proportional to the amount yet to be learned. Let \(y=f(t)\) be the amount of information learned up to time \(t .\) Construct and solve a differential equation that is satisfied by \(f(t).\)
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