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Chapter 10: Problem 9

Find a constant solution of \(y^{\prime}=t^{2} y-5 t^{2}\).

Short Answer

Expert verified
The constant solution is \(y = 5\).

Step by step solution

01

Identify the conditions for a constant solution

To find a constant solution, realize that a constant function does not change with respect to the independent variable. Therefore, the derivative of a constant function is zero. Set the left side of the differential equation to zero: \(y' = 0\).
02

Set up the equation

Given the differential equation \(y^{\text{'}} = t^{2} y - 5 t^{2}\), and knowing that \( y^{\text{'}} = 0\), set the equation to: \(0 = t^{2} y - 5 t^{2}\).
03

Solve for y

Isolate \(y\) by adding \(5t^2\) to both sides and then dividing by \(t^2\): \(0 + 5t^2 = t^2 y - 5t^2 + 5t^2\). This simplifies to \(5t^2 = t^2 y\), and then divide both sides by \(t^2\) to obtain \(y = 5\).

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Key Concepts

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

constant solution
A constant solution in differential equations is a special type of solution where the function remains unchanged no matter what the independent variable is. For example, if we're dealing with the differential equation given as \(y^{\prime}=t^{2} y-5 t^{2}\), we need to find a solution where the value of \(y\) stays the same regardless of \(t\).
To start, assume that the derivative of such a constant solution is zero. Hence, we can say \(y^{\prime} = 0\).
Next, substitute this back into the differential equation to isolate the constant value of \(y\).
Remember these steps to identify a constant solution:
  • Assume the derivative is zero since for any constant, its rate of change (derivative) is zero.
  • Set the left side of the equation to zero and solve for the constant value.
derivative
The derivative is a measure of how a function changes as its input changes. In mathematical terms, for a function \(y(t)\), the derivative \(y'\) gives the rate at which \(y\) changes with respect to \(t\).
When we say \(y'\), we are essentially calculating the slope of the tangent to the curve defined by the function at any point.
For instance, in our differential equation \(y^{\prime}=t^{2} y-5 t^{2}\), \(y'\) tells us how the function \(y\) changes as \(t\) changes.
  • The first step is to set \(y' = 0\) for a constant solution.
  • Next, substitute this value into the equation and solve for \(y\) to find the constant solution.
The concept of the derivative is crucial for understanding how functions evolve and is fundamental in calculus.
isolate variable
Isolating a variable means to manipulate an equation in such a way that you get the variable alone on one side of the equation. This is a crucial step in solving most algebraic and differential equations.
Let's break it down using our differential equation example: \(y' = t^{2} y - 5 t^{2}\).
Since we're seeking a constant solution, we have already set \(y' = 0\). This simplifies our equation to: \(0 = t^{2} y - 5 t^{2}\).
  • Add \(5t^2\) to both sides: \(0 + 5t^2 = t^2 y - 5t^2 + 5t^2\).
  • Simplify to get: \(5t^2 = t^2 y\).
  • Finally, divide both sides by \(t^2\): \(5 = y\).
Now, we have isolated \(y\) and found it equals 5. Simple logical steps like these make complex solutions more manageable.

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Most popular questions from this chapter

Suppose that the Consumer Products Safety Commission issues new regulations that affect the toy-manufacturing industry. Every toy manufacturer will have to make certain changes in its manufacturing process. Let \(f(t)\) be the fraction of manufacturers that have complied with the regulations within \(t\) months. Note that \(0 \leq f(t) \leq 1 .\) Suppose that the rate at which new companies comply with the regulations is proportional to the fraction of companies who have not yet complied, with constant of proportionality \(k=.1\). (a) Construct a differential equation satisfied by \(f(t).\) (b) Use Euler's method with \(n=3\) to estimate the fraction of companies that comply with the regulations within the first 3 months. (c) Solve the differential equation in part (a) and compute \(f(3).\) (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.

Review concepts that are important in this section. In each exercise, sketch the graph of a function with the stated properties. Domain: \(0 \leq t \leq 8 ;(0,6)\) is on the graph; the slope is always negative, the slope becomes more negative as \(t\) increases from 0 to \(3,\) and the slope becomes less negative as \(t\) increases from 3 to 8.

Solve the following differential equations with the given initial conditions. $$y^{\prime}=2 t e^{-2 y}-e^{-2 y}, y(0)=3$$

Solve the following differential equations with the given initial conditions. $$\frac{d y}{d t}=\left(\frac{1+t}{1+y}\right)^{2}, y(0)=2$$
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